Soliton solutions for a class of critical Schrödinger equations with Stein–Weiss convolution parts in $\mathbb {R}^2$

Abstract

We consider the following class of quasilinear Schrödinger equations introduced in plasma physics and nonlinear optics with Stein–Weiss convolution parts

$$\begin{aligned} -\Delta u+V(x) u+\frac{\kappa }{2} u\Delta (u^2)=\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^2}\frac{H(u)}{|x-y|^\mu |y|^\beta }dy\Bigg ) h(u),~x\in \mathbb {R}^2, \end{aligned}$$

where $\kappa \in \mathbb {R}\backslash \{0\}$ is a parameter, $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$ and H is the primitive of h that fulfills the critical exponential growth in the Trudinger–Moser sense. For $\kappa <0$: (i) via using a change of variable argument and the mountain-pass theorem, we investigate the existence of ground state solutions only assuming that $V\in C^0(\mathbb {R}^2,\mathbb {R}^+)$ and $\inf _{x \in \mathbb {R}^2}V(x)>0$, which complements and generalizes the problems proposed in our recent work in Alves and Shen (J Differ Equ 344:352–404, 2023); (ii) by developing a new type of Trudinger–Moser inequality, we establish a Pohoz̆aev type ground solution by the constraint minimization approach when $V\equiv 1$. Moreover, if $\kappa >0$ is small, combining the mountain-pass theorem and Nash–Moser iteration procedure, we obtain the existence of nontrivial solutions, where the asymptotical behavior is also considered when $\kappa \rightarrow 0^+$. It seems that the results presented above are even new for the case $\kappa =0$.

1 Introduction and main results

In this paper, we are interested in the existence of nontrivial solutions for the nonlocal Schrödinger equations introduced in plasma physics and nonlinear optics with Stein–Weiss convolution parts

$$\begin{aligned} -\Delta u+ V(x)u+\frac{\kappa }{2} u\Delta (u^2)=\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^2}\frac{H(u)}{|x-y|^\mu |y|^\beta }dy\Bigg ) h(u),~x\in \mathbb {R}^2,\qquad \end{aligned}$$

(1.1)

where $\kappa \in \mathbb {R}\backslash \{0\}$ is a parameter, $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$ and H is the primitive of h that fulfills the critical exponential growth in the Trudinger–Moser sense.

If $\kappa =0$ in Eq. (1.1), it belongs to the so-called Schrödinger equations with Stein–Weiss convolution parts

$$\begin{aligned} -\Delta u+ V(x)u =\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^2}\frac{H(u)}{|x-y|^\mu |y|^\beta }dy\Bigg ) h(u),~x\in \mathbb {R}^2, \end{aligned}$$

(1.2)

where $\beta >0$ and $0<\mu <2$ and $0<2\beta +\mu <N$. To treat Eq. (1.2) variationally, the Stein–Weiss inequality which can also be known as the weighted Hardy–Littlewood–Sobolev inequality (HLS in short) usually plays an important role. We recall it below

Proposition 1.1

(see e.g. [55]) Suppose that $r,s> 1$, $0<\mu <N$, $\overline{\beta }+\beta \ge 0$ and $\overline{\beta }+\beta +\mu \le N$, $\varphi \in L^r(\mathbb {R}^N)$ and $\psi \in L^s(\mathbb {R}^N)$. There is a sharp constant $C=C(\overline{\beta },\beta ,\mu ,N,s,r)>0$, independent of $\varphi$ and $\psi$, such that

$$\begin{aligned} \int _{\mathbb {R}^N}\int _{\mathbb {R}^N}\frac{|\varphi (x)||\psi (y)|}{|x|^{\overline{\beta }} |x-y|^{\mu }|y|^{\beta }} dxdy\le C|\varphi |_r|\psi |_s, \end{aligned}$$

(1.3)

where

$$\begin{aligned} 1-\frac{1}{r}-\frac{\mu }{N}< \frac{\overline{\beta }}{N}<1-\frac{1}{r}\quad \text {and}\quad \frac{1}{r}+\frac{1}{s}+\frac{\overline{\beta }+\beta +\mu }{N}=2. \end{aligned}$$

If we suppose that $\varphi (x)=\psi (x)=|u(x)|^p$ in the weighted Hardy–Littlewood–Sobolev inequality (1.3) with $\overline{\beta }=\beta$ and $s=r$ as well as $2\beta +\mu \le N$, then the integral

$$\begin{aligned} \int _{\mathbb {R}^N}\int _{\mathbb {R}^N}\frac{|u(x)|^p|u(y)|^p}{|x|^{ \beta } |x-y|^{\mu }|y|^{\beta }} dxdy \end{aligned}$$

is well-defined provided

$$\begin{aligned} \frac{N-2}{2N-2\beta -\mu }\le \frac{1}{p}\le \frac{N}{2N-2\beta -\mu }. \end{aligned}$$

(1.4)

As pointed out in some previous papers, e.g. [9, 21], for $N\ge 3$, one could regard $2^*_{\beta ,\mu }=\frac{2N-2\beta -\mu }{N-2}$ and $2_{*,\beta ,\mu }=\frac{2N-2\beta -\mu }{N}$ the upper and lower critical Sobolev exponents, respectively. In fact, the critical Sobolev exponents are driven by the following two inequalities

$$\begin{aligned} \Bigg (\int _{\mathbb {R}^N}\int _{\mathbb {R}^N}\frac{|u(x)|^{2^*_{\beta ,\mu }}|u(y)|^{2^*_{\beta ,\mu }}}{|x|^{ \beta } |x-y|^{\mu }|y|^{\beta }} dxdy\Bigg )^{ \frac{1}{2^*_{\beta ,\mu }}}\le C^*\int _{\mathbb {R}^N}|\nabla u|^2dx, \end{aligned}$$

and

$$\begin{aligned} \Bigg (\int _{\mathbb {R}^N}\int _{\mathbb {R}^N}\frac{|u(x)|^{2_{*,\beta ,\mu }}|u(y)|^{2_{*,\beta ,\mu }}}{|x|^{ \beta } |x-y|^{\mu }|y|^{\beta }} dxdy\Bigg )^{ \frac{1}{2_{*,\beta ,\mu }}}\le C_*\int _{\mathbb {R}^N}| u|^2dx. \end{aligned}$$

Very recently, by introducing a nonlocal version of the concentration-compactness principle, Du et al. [21] considered the existence of nontrivial solutions and investigated the regularity, symmetry of positive solutions by the moving plane arguments for the equation

$$\begin{aligned} -\Delta u =\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^N}\frac{|u(y)|^{2^*_{\beta ,\mu }}}{|x-y|^\mu |y|^\beta }dy\Bigg ) |u|^{2^*_{\beta ,\mu }-2}u,~x\in \mathbb {R}^N. \end{aligned}$$

By developing the Pohoz̆aev identity of

$$\begin{aligned} -\Delta u +u=\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^N}\frac{|u(y)|^{p}}{|x-y|^\mu |y|^\beta }dy\Bigg ) |u|^{p-2}u,~x\in \mathbb {R}^N, \end{aligned}$$

they also studied the existence and non-existence nontrivial solutions with p satisfying (1.4). Moreover, Yang et al. [58] investigated the symmetry, regularity and asymptotical properties as well as sufficient conditions for the nonexistence of nontrivial solutions of semilinear elliptic systems

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u =\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^N}\frac{|v(y)|^{p}}{|x-y|^\mu |y|^\beta }dy\Bigg ) |u|^q, &{}\quad x\in \mathbb {R}^N, \\ -\Delta v=\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^N}\frac{|u(y)|^{p}}{|x-y|^\mu |y|^\beta }dy\Bigg ) |v|^q, &{}\quad x\in \mathbb {R}^N, \end{array} \right. \end{aligned}$$

where $N\ge 3$ and $p,q>1$. After the aforementioned works, under the assumptions

$(\overline{V}_0)$:

$V\in C^0(\mathbb {R}^N)$ and $\inf _{x\in \mathbb {R}^N}V(x)>0$;

$(\overline{V}_1)$:

there exists a constant $M_0>0$ such that the set $\{x\in \mathbb {R}^N:V(x)\le M_0\}$ has a finite Lebesgue measure and $\Omega = V^{-1}(0)$ is a non-empty set,

Zhang–Tang [61] especially explored the existence and concentration of solutions for the equation

$$\begin{aligned} -\Delta u +\lambda V(x)u=\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^N}\frac{|u(y)|^{p}}{|x-y|^\mu |y|^\beta }dy\Bigg ) |u|^{p-2}u,~x\in \mathbb {R}^N, \end{aligned}$$

when $\lambda >0$ is sufficiently large. It is widely known that the potential V with assumptions $(\overline{V}_0)-(\overline{V}_1)$ would be denoted by the steep potential well, see e.g. [13]. In a latest paper [9], by considering Eq. (1.2) with $V(x)\equiv 1$ and $N=2$, Alves-Shen combined the mountain-pass theorem and Pohoz̆aev identity to establish the existence of nontrivial solutions, mountaion-pass type solutions, least energy solutions and ground state solutions in the radially symmetric space, where they also studied Eq. (1.2) under the conditions $(\overline{V}_0)$ and

$(\overline{V}_2)$:

$V (x)\rightarrow \infty$ as $|x|\rightarrow \infty$; or more generally $|\{x\in \mathbb {R}^2:V(x)\le M\}|<\infty$ for every $M>0$; or the function $[V(x)]^{-1}$ belongs to $L^1(\mathbb {R}^2)$.

The potential V satisfying $(\overline{V}_2)$ is known to be coercive, ses e.g. [4, 19, 44]. In summary, to restore the compactness of Eq. (1.2) caused by the whole Euclidean space $\mathbb {R}^N$ and the nonlinearity involving critical growth, the usual idea is to suppose $(\overline{V}_1)$, or V being radially symmetric, or $(\overline{V}_2)$. Therefore, one may naturally ask the following two Questions:

(I):

Could we investigate the existence results for Eq. (1.2) by assuming that the positive potential V is periodic?

(II):

Could we even establish the very same results in Question (1) without the periodic assumption on V? In other words, whether we can get the existence results for Eq. (1.2) just by assuming

$$\begin{aligned} V\in C^0(\mathbb {R}^2) \quad \text{ and } \quad \inf _{x\in \mathbb {R}^2}V(x)>0. \end{aligned}$$

(V)

However, as mentioned in [9, Remark 1.5], the variational functional (even for the periodic potential V) corresponding to Eq. (1.2) does never remain translation invariance in general, which brings a lot of difficulties to solve the problem for a large class of potential V. In our opinion, the Questions (I) and (II) above are mathematically interesting and one of the main purposes in this paper is to explore them. Concerning some other interesting results for the Schrödinger equations with Stein–Weiss convolution parts, we refer to [28, 59, 61, 62] and the references therein.

Let $\beta =0$ in Eq. (1.2) and $|x|^{-\mu }$ can be reviewed as the classic Riesz potential, then the nonlinear Schrödinger equation

$$\begin{aligned} -\Delta u+ u=(|x|^{-\mu }*H(u))h(u), ~ x\in \mathbb {R}^N, \end{aligned}$$

(1.5)

is closely related to the Choquard equation arising from the study of Bose–Einstein condensation and can be exploited to describe the finite-range many-body interactions between particles, where $*$ denotes a convolution operator. With respect to the relevant physical case in which $N = 3$, $\mu =1$ and $H(u)=u^2$, Eq. (1.5) turns into the Choquard-Pekar equation which was introduced by Pekar [46] to describe a polaron at rest in the quantum field theory. In [31], Choquard exploited this equation to characterization an electron trapped in its own hole as an approximation to the Hartree-Fock theory for a one component plasma. After a while, by means of the variational methods, Lieb [30] and Lions [33] obtained the existence and uniqueness of positive solutions to (1.5). The regularity, radial symmetry and decay property of the ground state solution were considered in [38, 42]. Equation (1.5) and its variants have received many attentions by many mathematicians because of the appearance of the convolution type nonlinearities over the past decades. We should refer the reader to [1, 7, 8, 42, 53] and the references therein, particular by [43], for a very abundant and meaningful review of the Choquard equations. In fact, Moroz et al. in [41] proposed Eq. (1.5) to be a model for self-gravitating particles in the context as it can be regarded as the Schrödinger–Newton equation.

Then, we would introduce some results on Eq. (1.1) with $\kappa \ne 0$. Solutions of like it are usually used to the search of certain kinds of standing wave solutions to the nonlinear Schrödinger equation

$$\begin{aligned} i\partial \psi _t=-\Delta \psi +W(x)\psi -\eta (|\psi |^2)\psi +\frac{\kappa }{2}[\Delta \rho (|\psi |^2)]\rho ^\prime (|\psi |^2)\psi , \end{aligned}$$

(1.6)

where $\psi : \mathbb {R}^2 \times \mathbb {R}\rightarrow \mathbb {C}$, $\kappa \in \mathbb {R}\backslash \{0\}$, $W: \mathbb {R}^2\rightarrow \mathbb {R}$ is a given potential and $\eta : \mathbb {R}^+ \rightarrow \mathbb {R}$ and $\rho : \mathbb {R}^+ \rightarrow \mathbb {R}$ are appropriate functions. Based on several types of nonlinear term $\rho (s)$, there are a rich researcher topic in some areas of physics, see e.g. [48]. In the present paper, motivated by [26], we would mainly focus on the superfluid film equation in plasma physics, which corresponds to the case $\rho (s)=s$. In the meanwhile, it is worthy mentioning here that the scillating soliton instabilities during microwave, laser heating of plasma and so on appeared in nonlinear optics, see [24, 49] for instance.

Due to the real physical applications on Eq. (1.6), there are extensive bibliographies in the study of it by variational methods. To deal with the problem variationally, except in dimension one like [48], the principal barrier is to determine a suitable work space in which the associated energy functional is well-defined and of $C^1$-class. By introducing a suitable metric space, authors in [37, 51] exploited the constrain manifold of Nehari type and Nehari–Pohoz̆aev type to study (1.6), respectively. On the other hand, the perturbation procedure developed in [35] was utilized to investigate the quasilinear Schrödinger equations. Besides, there exists another way, namely a change of variable, to overcome the difficult mentioned early. To explain it clearly, we shall split into two cases depending on whether the parameter $\kappa$ is positive or negative.

With respect to the case in which the parameter $\kappa <0$, Liu et al. [36] performed the change of variable: $v=f^{-1}(u)$, where f(t) is defined by

$$\begin{aligned} f^\prime (t)=\frac{1}{\sqrt{1-\kappa f^2(t) }}~\text {on}~[0,+\infty )~\text {and}~f(t)=-f(-t)~\text {on}~(-\infty ,0]. \end{aligned}$$

(1.7)

By this change of variable in (1.7), they transformed the quasilinear equation $-\Delta u+V(x)+\frac{\kappa }{2}u\Delta (u^2)=h(u)$ into the semilinear one $-\Delta v=f^\prime (v)[h(f(v))-V(x)f(v)]$. Subsequently, with this change of variable to deal with the case $\kappa <0$, see e.g. [11, 12, 17, 20, 54] ant the references therein.

Obviously, the transformation in (1.7) is unapplicable when $\kappa >0$. So, it seems that the studies in this direction on the existence results are not as fruitful as the case $\kappa <0$. In [10], by introducing a new type change of variable, with the help of the mountain-pass theorem and Nash–Moser iteration technique, the authors established the existence of nontrivial solutions when the parameter $\kappa >0$ is sufficiently small. Along this line, there exist some similar results on quasiliear Schrödinger equations by using this argument, see e.g. [25, 45, 52]. It should be noted that, for $N\ge 3$, the Sobolev critical exponents are $22^*=4N/(N-2)$ and $2^*$ when $\kappa <0$ and $\kappa >0$, respectively, see [45] for more details.

Now, we shall turn to the main topics in this paper. The other aim in this work is to consider the existence results for Eq. (1.1) with critical exponential growth by exploiting the variational methods. Motivated by [45] as well as the Trudinger–Moser type inequality, for the cases $\kappa <0$ and $\kappa >0$, one can say that a function h possesses critical exponential growth if there is a constant $\alpha _{0}>0$ such that

$$\begin{aligned} \lim \limits _{|t|\rightarrow +\infty } \frac{|h(t)|}{e^{\alpha t^{4}}}= \left\{ \begin{array}{ll} 0, &{}\quad \forall \alpha >\alpha _{0}, \\ +\infty , &{}\quad \forall \alpha <\alpha _{0}, \end{array} \right. \end{aligned}$$

(1.8)

and

$$\begin{aligned} \lim \limits _{|t|\rightarrow +\infty } \frac{|h(t)|}{e^{\alpha t^{2}}}= \left\{ \begin{array}{ll} 0, &{}\quad \forall \alpha >\alpha _{0}, \\ +\infty , &{}\quad \forall \alpha <\alpha _{0}, \end{array} \right. \end{aligned}$$

(1.9)

respectively. This definitions can be found in some literatures, see e.g. [3, 22].

To the best knowledge of us, there exist very few results on Schrödinger equations with Stein–Weiss convolution parts. As explained in [9], the imbedding $H_0^1(\Omega )\hookrightarrow L^p(\Omega )$ with $1\le p<+\infty$ can never guarantee $H_0^1(\Omega )\hookrightarrow L^\infty (\Omega )$ for bounded domain $\Omega \subset \mathbb {R}^2$. So, one is led to ask if there is another kind of maximal growth in this situation. Indeed, the authors in [40, 47, 56] particularly established the following sharp maximal exponential integrability for functions in $H_{0}^{1}(\Omega )$:

$$\begin{aligned} \sup \limits _{u\in H_{0}^{1}(\Omega ):\Vert \nabla u\Vert _{L^2 (\Omega )}\le 1}\int _{\Omega }e^{\alpha u^{2}}dx\le C|\Omega |~\text{ if }~\alpha \le 4\pi , \end{aligned}$$

where the constant $C=C(\alpha )>0$ and $|\Omega |$ stands for the Lebesgue measure of $\Omega$.

There are a lot of generalizations to the well-known Trudinger–Moser inequality in many directions. For example, Adimurthi and Sandeep proved in [5] that the inequality

$$\begin{aligned} \sup \limits _{u\in H_{0}^{1}(\Omega ):\Vert \nabla u\Vert _{L^2 (\Omega )}\le 1}\int _{\Omega }|x|^{-s}e^{\alpha u^{2}}dx<+\infty , \end{aligned}$$

holds if and only if $\frac{\alpha }{4\pi }+\frac{s}{2}\le 1$, where $\alpha >0$ and $s\in [0,2)$.

Because of the appearance of the singular weight $|x|^{-\beta }$ in Eq. (1.1), we prefer to use the following version of the Trudinger–Moser inequality in the whole Euclidian space $\mathbb {R}^2$ established by Adimurthi and Yang [4].

Proposition 1.2

For all $\alpha >0$, $s\in [0,2)$ and $u\in H^1(\mathbb {R}^2)$, there holds

$$\begin{aligned} |x|^{-s}(e^{\alpha u^2}-1)\in L^1(\mathbb {R}^2). \end{aligned}$$

(1.10)

Moreover, for each $r \in (0,1)$ and $M>0$, there exists a universal constant $C=C(r,M)>0$, independent of u, such that

$$\begin{aligned} \sup _{u \in \mathcal {M}_{r,M}}\int _{\mathbb {R}^2}|x|^{-s}(e^{\alpha u^2}-1)dx\le C \end{aligned}$$

(1.11)

if and only if $\frac{\alpha }{4\pi }+\frac{s}{2}\le 1$, where

$$\begin{aligned} \mathcal {M}_{r,M}=\{u \in H^{1}(\mathbb {R}^2)\,:\, |\nabla u|_{2}\le r \quad \text{ and } \quad |u|_{2}\le M \}. \end{aligned}$$

As to some other generalizations, extensions and applications of the Trudinger–Moser inequalities for bounded and unbounded domains, we refer to [22, 29] and the references therein.

Before stating the existence results briefly in the present paper, we shall introduce some notations and definitions. Throughout this paper, by the assumption (V), one can easily find that

$$\begin{aligned} E\triangleq \Bigg \{u\in H^1(\mathbb {R}^2):\int _{\mathbb {R}^2}V(x)|u|^2dx<+\infty \Bigg \} \end{aligned}$$

is a Hilbert space equipped with the inner product and norm

$$\begin{aligned} (u,v)=\int _{\mathbb {R}^2}\big [\nabla u\nabla v+ V(x)uv\big ]dx~ \text {and}~ \Vert u\Vert =(u,u)^{\frac{1}{2}}, ~\forall u,v\in E. \end{aligned}$$

Obviously, the imbedding $E\hookrightarrow H^1(\mathbb {R}^2)$ is continuous, where $H^1(\mathbb {R}^2)$ admits the usual inner product and norm. Let $L^q(\mathbb {R}^2)$ ($1\le q\le \infty$) be the usual Lebesgue space with standard norm $|u|_q$. Moreover, for all $s\in [0,2)$, we need the weighted Lebesgue space $L^q(\mathbb {R}^2,|x|^{-s}dx)$ with $q\in [1,+\infty )$ defined by

$$\begin{aligned} L^q(\mathbb {R}^2,|x|^{-s}dx)\triangleq \Bigg \{u\in L^q(\mathbb {R}^2):\int _{\mathbb {R}^2}|x|^{-s}|u|^qdx<+\infty \Bigg \}. \end{aligned}$$

We denote by C and $C_i$ ($i=1, 2,\cdots$) for various positive constants whose exact value may change from lines to lines but are not essential to the analysis of the problem. Let $(X,\Vert \cdot \Vert _X)$ be a Banach space with dual space $(X^{-1},\Vert \cdot \Vert _{X^{-1}})$, and $\Phi$ be functional on X. The Cerami sequence at a level $c\in \mathbb {R}$ ($(C)_c$ sequence in short) corresponding to $\Phi$ means that $\Phi (x_n)\rightarrow c$ and $(1+\Vert x_n\Vert _X)\Vert \Phi ^{\prime }(x_n)\Vert _{X^{-1}}\rightarrow 0$ as $n\rightarrow \infty$, where $\{x_n\}\subset X$.

Firstly, we study the existence results for Eq. (1.1) in the case $\kappa <0$ under the condition (V):

(V):

$V\in C^0(\mathbb {R}^2)$ and $\inf _{x\in \mathbb {R}^2}V(x)>0$.

Moreover, we suppose that the nonlinearity h satisfies (1.8) and the following hypotheses:

$(h_1)$:

$h\in C(\mathbb {R})$, $h(s)\equiv 0$ for all $s\le 0$ and $h(s)=o(s^{\frac{2-\mu }{2}})$;

$(h_2)$:

there exist some constants $s_0>0$, $M_0>0$ and $\vartheta \in (0,1]$ such that

$$\begin{aligned} 0<s^{\vartheta }H(s)\le M_0h(s),~\forall s\ge s_0; \end{aligned}$$

$(h_3)$:

$\displaystyle \liminf _{s\rightarrow +\infty }H(s)/e^{\alpha _0s^4}\triangleq \beta _0>0$,

$(h_4)$:

$h\in C^1(\mathbb {R})$ and there exists a constant $\delta \in [\frac{1}{2},1)$ such that $H(s)h^\prime (s)\ge \delta h^2(s)$ for all $s\ge 0$.

We would like to highlight that a great many functions h satisfying $(h_1)-(h_4)$, for example,

$$\begin{aligned} h(s)=|s|^{p-2}s(e^{\alpha _0 s^4}-1),~\forall s\in \mathbb {R}~\text {with some} ~p>2. \end{aligned}$$

Because $\kappa <0$ in Eq. (1.1) is arbitrary, without loss of generality, in the sequel we shall always suppose that $\kappa \equiv -1$ throughout this paper for simplicity. We obtain the following result.

Theorem 1.3

Let (V) and (1.8) with $(h_1)-(h_4)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, then Eq. (1.1) admits at least a nontrivial solution $u_0\in X$ (see Sect. 2 below). If $\delta =\frac{1}{2}$ in $(h_4)$, then $\mathcal {J}_f(v_0) = \inf _{v\in \mathcal {N}_f} \mathcal {J}_f(v)$, where $v_0=f^{-1}(u_0)$ as well as f given in (1.7) and the Nehari manifold $\mathcal {N}_f\triangleq \{v\in X\backslash \{0\}:\mathcal {J}^\prime _f(v)[v]=0\}$ with $\mathcal {J}_f:X\rightarrow \mathbb {R}$ defined by

$$\begin{aligned} \mathcal {J}_f(v)=\frac{1}{2}\int _{\mathbb {R}^2} |\nabla v|^2+V(x)f^2(v)]dx-\frac{1}{2} \int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(f(v(x)))H(f(v(y)))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx. \end{aligned}$$

Remark 1.4

We have to point out that the condition, like $(h_4)$, was originally proposed by Cassani and Tarsi in [16], later generalized by Shen, Rădulescu and Yang in [53]. Whereas, it is not obvious to certify that every $(C)_c$ sequence of $\mathcal {J}_f$ is uniformly bounded because of the appearance of quasilinear term. In particular, compared with [53, Lemma 3.6], we must introduce some new techniques to handle the case $\delta =\frac{1}{2}$. Moreover, although it fails to cover the case $\delta \in (0,\frac{1}{2})$, we do not hesitate to confirm that this interval is optimal when the potential V just belongs to $C^0(\mathbb {R}^2)$. Finally, we succeed in deriving the affirmative answers to the Questions (I) and (II) presented above.

Remark 1.5

Proceeding as some similar arguments in [53, Remark 1.6], we shall also verify some key observations on h and H as follows. On the one hand, by $(h_4)$, one could conclude that $h^\prime (s)\ge 0$ for all $s>0$ and then h is nondecreasing on $s\in (0,+\infty )$ which implies that

$$\begin{aligned} 0<H(s)=\int _{0}^{s}h(t)dt\le h(s)s,~\forall s>0. \end{aligned}$$

(1.12)

On the other hand, one would conclude that $(H(s)/h(s))^\prime \le 1-\delta$ for each $s>0$ by $(h_4)$. This combined with the fact that $0<H(s)=\int _{0}^{s}h(t)dt$ yields

$$\begin{aligned} 0<H(s) \le (1-\delta ) h(s)s,~\forall s>0. \end{aligned}$$

(1.13)

In particular, by $(h_4)$ and (1.13), we could immediately have that

$$\begin{aligned} (1-\delta )h^\prime (s)s\ge \delta h(s),~\forall s>0. \end{aligned}$$

(1.14)

Let us recall that the imbedding $E\hookrightarrow H^1(\mathbb {R}^2)$ is continuous, then $E\hookrightarrow L^q(\mathbb {R}^2,|x|^{-\beta }dx)$ is compact since $H^1(\mathbb {R}^2)\hookrightarrow L^q(\mathbb {R}^2,|x|^{-\beta }dx)$ is compact for every $q\ge 2$, see Lemma 2.3 below in detail. Thereby, inspired by [9, Theorem 5.8], we suppose that the nonlinearity h satisfies the conditions

$(H_1)$:

For $\alpha _0>0$ given by (1.8), there exist constants $b_1,b_2>0$ such that for all $s\in \mathbb {R}^+$,

$$\begin{aligned} 0<h(s)\le b_1|s|^{\frac{2 -\mu }{2}}+b_2(e^{\alpha _0 s^4}-1); \end{aligned}$$

$(H_2)$:

$H(s)\le C_0|s|^{\frac{4 -\mu }{2}}+C_0h(s)$ for all $s\in \mathbb {R}^+$;

$(H_3)$:

$\tilde{H}(s)\le \tilde{H}(t)$ for all $0<s<t$, where $\tilde{H}(s)=h(s)s-2H(s)$;

$(H_4)$:

$\displaystyle \lim _{|u|\rightarrow \infty }H(s)/|s|^2=+\infty$ in $x\in \mathbb {R}^2$;

$(H_5)$:

$\displaystyle \liminf _{s\rightarrow +\infty }sh( s)H( s)e^{-2\alpha _0s^4} \ge \overline{\beta }_0>\displaystyle \inf _{\rho >0}\frac{e^{\frac{4-2\beta -\mu }{4}}V_\rho \rho ^2}{\rho ^{4-2\beta -\mu }\alpha _0^2} \frac{(4-\mu )^2}{(2-\mu )(3-\mu )}$, where $V_\rho \triangleq \displaystyle \sup _{|x|\le \rho }V(x)$.

As a consequence, we can prove the following result whose proof is omitted.

Corollary 1.6

Let (V) and $(H_1)-(H_5)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, then Eq. (1.1) has at least a nontrivial solution in X.

Motivated by [37, 51], we investigate the existence of Pohožaev type ground state solutions for Eq. (1.1) with $V\equiv 1$ and $\kappa =-1$, that is,

$$\begin{aligned} -\Delta u+ u-\frac{1}{2} u\Delta (u^2)=\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^2}\frac{H(u)}{|x-y|^\mu |y|^\beta }dy\Bigg ) h(u),~x\in \mathbb {R}^2. \end{aligned}$$

(1.15)

To the end, instead of $(h_2)-(h_3)$, we need to suppose that h satisfies the following condition

$(h_5)$:

there exist two constants $p>1$ and a sufficiently large $\xi >0$ (whose lower bounded can be determined later, see e.g. Theorem 1.8 below) such that $H(s)\ge \xi s^{p}$ for all $s\in [0,+\infty )$.

Thus, the second main result in this paper can be stated as follows.

Theorem 1.7

Suppose that h satisfies (1.8), $(h_1)$ and $(h_5)$. If $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, then Eq. (1.15) admits a ground state solution $u\in \mathcal {X}$ satisfying

$$\begin{aligned} I(\overline{u})=\inf _{u\in \mathcal {X}\backslash \{0\}}\max _{t>0}I(u( t^{-1}x )),~\text {with}~ \mathcal {X}\triangleq \{u\in H^1(\mathbb {R}^2):u^2\in H^1(\mathbb {R}^2)\}, \end{aligned}$$

where the corresponding variational $I:\mathcal {X}\rightarrow \mathbb {R}$ is defined by

$$\begin{aligned} I(u)= & {} \frac{1}{2}\int _{\mathbb {R}^2}[(1+ u^2)|\nabla u|^2+u^2]dx\nonumber \\{} & {} -\frac{1}{2} \int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(u(x))H(u(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx. \end{aligned}$$

(1.16)

For the work space $\mathcal {X}$ dealt with Eq. (1.15), we would equip it with the following distance

$$\begin{aligned} d_{\mathcal {X}}(u,v)\triangleq \frac{1}{2}\sqrt{4\Vert u-v\Vert _{H^1(\mathbb {R}^2)}^2+ |\nabla u^2-\nabla v^2|_2^2},~\forall u,v\in \mathcal {X}. \end{aligned}$$

Obviously, the distance is definitely different from those in [37, 51]. Moreover, $(\mathcal {X},d_{\mathcal {X}}(\cdot ,\cdot ))$ is a metric space rather than a vector space because it is not closed under the sum, but it is a complete metric space. Besides, it could be said that $u\in \mathcal {X}$ is a (weak) solution of Eq. (1.15) if there holds

$$\begin{aligned}{} & {} \int _{\mathbb {R}^2}[(1+ u^2)\nabla u\nabla \psi + (1+|\nabla u|^2) u\psi ]dx\\{} & {} \quad = \int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{h(u(x))\psi (x)H(u(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx,~\forall \psi \in C_0^\infty (\mathbb {R}^2). \end{aligned}$$

In a certain sense, the weak solutions of Eq. (1.15) are critical points of I. Combining the assumptions (1.8) and $(h_1)$ as well as Proposition 3.1 below, it can be verified that I is well-defined and of class $C^0(\mathcal {X})$. Moreover, since $u+\varphi \in \mathcal {X}$ for every $u\in \mathcal {X}$ and $\varphi \in C_{0}^\infty (\mathbb {R}^2)$, the Gateaux derivative of J could be computed as follows

$$\begin{aligned} I^\prime (u)[\varphi ]= & {} \int _{\mathbb {R}^2}[(1+ u^2)\nabla u\nabla \psi + (1+ |\nabla u|^2) u\psi ]dx\\{} & {} - \int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{h(u(x))\psi (x)H(u(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx. \end{aligned}$$

By Lemma 3.2 below, $u\in \mathcal {X}$ is a weak solution of (1.15) if and only if the Gateaux derivative of I in every direction $\varphi \in C_{0}^\infty (\mathbb {R}^2)$ vanishes. In view of Lemma 3.3, each weak solution $u\in \mathcal {X}\backslash \{0\}$ of (1.15) satisfies $u\in \mathcal {M}$, where

$$\begin{aligned} \mathcal {M}\triangleq & {} \Bigg \{u\in \mathcal {X}\backslash \{0\}: 2\int _{\mathbb {R}^2} |u|^2dx -(4-2\beta -\mu )\\{} & {} \int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(u(x))H(u(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx=0\Bigg \}. \end{aligned}$$

At this position, we shall present some explanations on Theorem 1.7 as follows:

• (1) Compared with [9, Theorem 1.4], the main contributions are threefold: (I) the quasilinear term which invokes the well-known Trudinger–Moser inequalities (1.10) and (1.11) in Proposition 1.2 to be unapplicable in Eq. (1.15) is involved; (II) the work space $\mathcal {X}$ does not need to be radially symmetric; (III) the constraints on h can be greatly relaxed.

• (2) In view of the condition $(h_1)$, it behaves as at the original point in the Choquard type setting, see e.g. [53]. As a consequence, to some extent, we shall also regard Eq. (1.15), or (1.1), as the Schrodinger equation of Choquard type with a singular nonlinearity. It is worthy mentioning here that the parameter $\beta \in (0,2)$ makes sure the imbedding $H^1(\mathbb {R}^2)\hookrightarrow L^q(\mathbb {R}^2,|x|^{-\beta }dx)$ is compact for every $q\ge 2$, see Sect. 2 below in detail. Hence, we cannot simply take it for granted that Theorem 1.3, Corollary 1.6 and Theorem 1.7 are also true for $\beta =0$.

• (3) In fact, combining the arguments adopted in [51] and Theorem 1.7, one can also treat Eq. (1.15) with a nonconstant potential after some slight modifications;

• (4) To recover the compactness, we fail to take the energy estimate by the assumptions $(h_2)-(h_3)$ since the Moser sequence functions $\{\overline{w}_n\}\subset \mathcal {X}$ for each fixed $n\in \mathbb {N}$, but $\int _{\mathbb {R}^2}|\overline{w}_n|^2|\nabla \overline{w}_n|^2dx\rightarrow +\infty$ as $n\rightarrow \infty$. Therefore, it would be interesting to construct a new type of Moser sequence functions to overcome this difficulty and we shall contemplate it in a further work.

Finally, we will concentrate on Eq. (1.1) for the case $\kappa >0$. As stated before, the maximal growth with respect to the nonlinearity h behaves like $e^{\alpha _0s^2}$ at infinity. So, this is one of essential differences from the case $\kappa <0$. To establish the existence results, we suppose that

$(h_6)$:

there exist two constants $\overline{p}>1$ and $\overline{\xi }>0$ such that $H(s)\ge \overline{\xi } s^{\overline{p}}$ for all $s\in [0,1]$;

$(h_7)$:

there exists a constant $\overline{\eta }>1$ such that $h(s)s\ge \overline{\eta } H(s)$ for all $s\ge 0$.

We then establish the third main result in this paper.

Theorem 1.8

Let (V) and (1.9) with $(h_1)$, $(h_6)-(h_7)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, if in addition $\displaystyle \lim _{s\rightarrow 0}h(s)/s=0$ and the constant $\overline{\xi }>0$ given by $(h_5)$ meets that $\overline{\xi }>\overline{\xi }_0$ as well as

$$\begin{aligned} \overline{\xi }_0\triangleq \max \Bigg \{\sqrt{\frac{3\overline{V}}{2^{\mu -5}\pi }}, \sqrt{\Bigg (\frac{36\alpha _0\overline{\eta }(\overline{p}-1) \pi \overline{V}}{\pi (4-2\beta -\mu )(\overline{\eta }-1)\overline{p}}\Bigg )^{ \overline{p}-1} \frac{3\overline{V}}{2^{\mu -5}\overline{p}\pi }}\Bigg \}, \end{aligned}$$

where $\overline{V}\triangleq [1+ \max _{x\in B_1(0)}V(x)]\in (0,+\infty )$, then there exists a constant $\kappa _0>0$ such that Eq. (1.1) possesses a nontrivial solution $u_\kappa \in E\cap L^\infty (\mathbb {R}^2)$ for all $\kappa \in (0,\kappa _0)$.

Remark 1.9

It is simple to observe that $(h_3)$ and $(h_4)$ are replaced with $(h_6)$ and $(h_7)$, respectively. We can never suppose $(h_6)$ to be that “$(h^\prime _5):\liminf _{s\rightarrow +\infty }H(s)e^{-\alpha _0 s^2}>0$", the biggest reason is that it is difficult to verify that Lemma 4.5 remains true if $(h^\prime _5)$ holds. Obviously, $(1-\delta )^{-1}>2$ in $(h_4)$ and $\eta >1$ in $(h_6)$, but it does not contradict with the depiction “the interval is optimal” in Remark 1.4. Moreover, we prefer to highlight here that the conditions $(h_1)$ and $(h_6)$ in Theorem 1.8 can be replaced with “$\lim _{s\rightarrow 0^+}H(s)/s^{\eta }\triangleq \overline{\xi }$" if $\overline{\xi }>0$ is suitably large. As a by-product of Theorem 1.8, the Questions-(II) mentioned above is fully accomplished to some extent.

Remark 1.10

To our best knowledge, both the results on $\kappa <0$ and $\kappa >0$ in Theorems 1.3, 1.7 and 1.8 are new for the Schrödinger equations with Stein–Weiss convolution parts and the nonlinearity involving critical exponential growth. Besides, we believe that our results may prompt further studies on these type equations.

In consideration of Theorem 1.8, the existence of nontrivial solutions of Eq. (1.1) heavily depends on the parameter $\kappa >0$ being sufficiently small, we wonder what will happen in Theorem 1.8 when $\kappa \rightarrow 0^+$. In other words, whether $\{u_\kappa \}$ converges in the sense of a subsequence and what does $\{u_\kappa \}$ converges to. For these purposes, we study the asymptotical behavior of $\{u_\kappa \}$ below.

Theorem 1.11

Under the assumptions in Theorem 1.8, let $u_\kappa \in H^1(\mathbb {R}^2)\cap L^\infty (\mathbb {R}^2)$ be a nontrivial solution of Eq. (1.1), then, passing to a subsequence if necessary, $u_\kappa \rightarrow u_0$ in $H^1(\mathbb {R}^2)$ as $\kappa \rightarrow 0^+$, where $u_0$ is a nontrivial solution of

$$\begin{aligned} -\Delta u+V(x)u=\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^2}\frac{H(u)}{|x-y|^\mu |y|^\beta }dy\Bigg ) h(u),~x\in \mathbb {R}^2. \end{aligned}$$

(1.17)

As far as we are concerned, it seems the first time to consider the asymptotical behavior of nontrivial solutions for the quasilinear Schrodinger equations in the case $\kappa >0$ with critical exponential growth.

The paper is organized as follows. By introducing some useful preliminaries which are important in the whole paper, we show the proof of Theorem 1.3 in Sect. 2. Sections 3 and 4 are devoted to the proofs of Theorems 1.7 and 1.8, 1.11, respectively.

2 Preliminaries and the proof of Theorem 1.3

In this section, we try to investigate the existence of nontrivial solutions of Eq. (1.1). Firstly, we observe that Eq. (1.1) is the Euler-Lagrange equation associated with the variational functional

$$\begin{aligned} J(u)= & {} \frac{1}{2}\int _{\mathbb {R}^2}[(1+ u^2)|\nabla u|^2+V(x)u^2]dx\\{} & {} -\frac{1}{2} \int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(u(x))H(u(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx,~\forall u\in E. \end{aligned}$$

According to the variational point of view, the starting obstacle with respect to Eq. (1.1) is to look for an appropriate function space where J is well-defined. To get though it, we shall make full use of the change of variables introduced in [36]. More precisely, we consider $v = f ^{-1}(u)$, where f is defined by (1.7) with $\kappa =-1$. Motivated by [17, 36], one can prove the following result.

Lemma 2.1

f given by (1.7) with $\kappa \equiv -1$ is an odd function and enjoys the following properties:

$(f_1)$:

$0<f^\prime (t)\le 1$ for all $t\in \mathbb {R}$;

$(f_2)$:

$|f(t)|\le |t|$ for all $t\in \mathbb {R}$;

$(f_3)$:

$|f(t)|\le 2^{1/4}|t|^{1/2}$ for all $t\in \mathbb {R}$;

$(f_4)$:

$f^\prime (0)=\lim _{t\rightarrow 0}f(t)/t=1$;

$(f_5)$:

$\lim _{t\rightarrow +\infty }f(t)/\sqrt{t}=2^{1/4}$;

$(f_6)$:

$\frac{1}{2}f^2(t)\le f(t)f^\prime (t)t\le f^2(t)$ for all $t\in \mathbb {R}$;

$(f_7)$:

there is a constant $C>0$ such that $|f(t)|\ge C|t|$ for $|t|\le 1$ and $|f(t)|\ge C\sqrt{|t|}$ for $|t|\ge 1$;

$(f_8)$:

there exist positive constants $C_1$ and $C_2$ satisfying

$$\begin{aligned} |t|\le C_1 |f(t)|+C_2|f(t)|^2,~\forall t\in \mathbb {R}; \end{aligned}$$

$(f_9)$:

$f^\prime (t)t$ is increasing and $f(t)f^\prime (t)t^{-1}$ is decreasing for all $t\in \mathbb {R}$.

After the change of variables $v = f ^{-1}(u)$, from J , we obtain the following functional $\mathcal {J}_f=J\circ f:E\rightarrow \mathbb {R}$ defined by

$$\begin{aligned} \mathcal {J}_f(v)=\frac{1}{2}\int _{\mathbb {R}^2}[|\nabla v|^2+V(x)|f(v)|^2]dx -\frac{1}{2}\int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(f(v(x)))H(f(v(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx \end{aligned}$$

which is well-defined in E and belongs to $C^1$ under the assumptions (V), (1.8) and $(h_1)$. Moreover, the critical points of $\mathcal {J}_f$ are weak solutions of the problem

$$\begin{aligned}{} & {} -\Delta v+ V(x)f(v)f^\prime (v)\nonumber \\{} & {} \quad =\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^2}\frac{H(f (v))}{|x-y|^\mu |y|^\beta }dy\Bigg ) h(f(v))f^\prime (v),~x\in \mathbb {R}^2. \end{aligned}$$

(2.1)

We observe that, if v is nonnegative, then u is nonnegative by $(h_1)$. Hence, to establish the existence of nontrivial solutions of (2.1), the critical point theorem introduced in [14, 39] will be applied.

Proposition 2.2

Let X be a Banach space and $\Phi \in C^1(X,\mathbb {R})$ Gateaux differentiable for all $v\in X$, with G-derivative $\Phi ^\prime (v)\in X^{-1}$ continuous from the norm topology of X to the weak $*$ topology of $X^{-1}$ and $\Phi (0) = 0$. Let S be a closed subset of X which disconnects (archwise) X. Let $v_0 = 0$ and $v_1\in X$ be points belonging to distinct connected components of $E\backslash X$. Suppose that

$$\begin{aligned} \inf _{S}\Phi \ge \varrho >0~\text {and}~\Phi (v_1)\le 0 \end{aligned}$$

and let $\Gamma =\{\gamma \in C([0,1],X):\gamma (0)~\text {and}~\gamma (1)=v_1\}$. Then

$$\begin{aligned} c=\inf _{\gamma \in \Gamma }\max _{t\in [0,1]}\Phi (\gamma (t))\ge \varrho >0 \end{aligned}$$

and there is a $(C)_c$ sequence for $\Phi$.

Before verifying that the functional $\mathcal {J}_f$ admits a mountain-pass geometry, instead of E introduced in Sect. 1, we need to choose a suitable work space. Motivated by [11, 12, 20], we define the space

$$\begin{aligned} X=\Bigg \{v\in H^{1}(\mathbb {R}^2):\int _{\mathbb {R}^2}V(x)|f(v)|^2dx<+\infty \Bigg \} \end{aligned}$$

endowed with the norm

$$\begin{aligned} \Vert v\Vert _X=| \nabla v|_2 +\inf _{\xi >0}\frac{1}{\xi }\Bigg [1+\int _{\mathbb {R}^2}V(x) |f(\xi v)|^2dx\Bigg ]. \end{aligned}$$

In particular, if $V(x)\equiv V_0>0$ for all $x\in \mathbb {R}^2$, the norm above is equivalent to the usual norm in $H^1(\mathbb {R}^2)$. By (V) and $(f_2)$, then $E\hookrightarrow X\hookrightarrow H^1(\mathbb {R}^2)$ is continuous which is crucial in this paper. Moreover, by the results in [20], the space $(X,\Vert \cdot \Vert _X)$ is a reflexive and Banach space and $C_0^\infty (\mathbb {R}^2)$ is dense in it.

Hereafter, we denote by $\Upsilon >0$ the best constant of the embedding $X\hookrightarrow H^1(\mathbb {R}^2)$, that is,

$$\begin{aligned} \Vert v\Vert _{H^1(\mathbb {R}^2)} \le \Upsilon \Vert v\Vert _X, ~\forall v \in X. \end{aligned}$$

(2.2)

Next, we prove an important embedding in our approach.

Lemma 2.3

If $q\ge 2$ and $s\in (0, 2)$, then the embedding $H^1(\mathbb {R}^2)\hookrightarrow L^q(\mathbb {R}^2,|x|^{-s}dx)$ is compact.

Proof

Combining the Egoroff’s theorem and Lebesgue’s Dominated Convergence theorem, the proof is standard and we refer the interested reader to [60, Theorem 1.2] for details. $\square$

Lemma 2.4

Let (V) and (1.8) with $(h_1)-(h_4)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, then the functional $\mathcal {J}_f$ possesses the following properties

(i):

there exist two constants $\varrho ,\rho >0$ such that $\mathcal {J}_f(v)\ge \varrho$ for all $v\in X$ with $\Vert v\Vert _X=\rho$;

(ii):

there exists a function $e\in X$ with $\Vert e\Vert _X\ge \rho$ such that $\mathcal {J}_f(e)<0$.

Proof

(i) Recalling (1.8) and $(h_1)$, for fixed $\alpha >\alpha _0$, $q\ge \frac{4-\mu }{2\nu }$ with $1/\nu +1/\nu ^\prime =1$ and for all $\varepsilon >0$

$$\begin{aligned} |H(s)|\le \varepsilon |s|^{\frac{4-\mu }{2}} +C(\alpha ,q,\varepsilon )|s|^{q}(e^{\alpha s^4}-1),~\forall s\in \mathbb {R}. \end{aligned}$$

(2.3)

Let $\varepsilon =1$ in (2.3) with suitable $\alpha$ and $\nu ^\prime$, taking $\Vert v\Vert _X^2<\frac{\pi (4-\mu -2\beta )}{2\Upsilon ^2\alpha _0}$, then we could apply the HLS inequality and (1.11) together with $(f_2)-(f_3)$ and Lemma 2.3 to derive

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(f(v(x)))H(f(v(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx \le C_\mu \Bigg (\int _{\mathbb {R}^2}\frac{|H(f(v))|^{\frac{4}{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{2}}\\&\quad \le C_\mu \Bigg (\int _{\mathbb {R}^2}\frac{ |v|^{2}}{|x|^{\frac{4\beta }{4-\mu }}}dx \Bigg )^{\frac{4-\mu }{2}} +C \Bigg (\int _{\mathbb {R}^2}|x|^{-\frac{4\beta }{4-\mu }}|v|^{\frac{4q}{4-\mu }}\left( e^{\frac{8\alpha }{4-\mu }|v|^2}-1\right) dx\Bigg )^{\frac{4-\mu }{2}}\\&\quad \le C \Vert v\Vert _X^{ 4-\mu } +C \Bigg (\int _{\mathbb {R}^2}|x|^{-\frac{4\beta }{4-\mu }}|v|^{\frac{4q\nu }{4-\mu }} dx\Bigg )^{\frac{4-\mu }{2\nu }} \Bigg (\int _{\mathbb {R}^2}|x|^{-\frac{4\beta }{4-\mu }}\\&\qquad \left( e^{\frac{8\alpha \nu ^\prime (1+\epsilon )^2 |\nabla v|_{2}^2}{4-\mu }\left( \frac{v}{(1+\epsilon )|\nabla v|_{2}}\right) ^2}-1\right) dx\Bigg )^{\frac{4-\mu }{2\nu ^\prime }}\\&\quad \le C \Vert v\Vert _X^{ 4-\mu }+C\Vert v\Vert _X^{2q}, \end{aligned} \end{aligned}$$

(2.4)

for $\epsilon \approx 0^+$. From this, for all $v\in X\backslash \{0\}$ with $\Vert v\Vert _X^2<\min \{{1},\frac{\pi (4-2\beta -\mu )}{2\Upsilon ^2\alpha _0}\}$, we obtain

$$\begin{aligned} \mathcal {J}_f(v)\ge \frac{1}{2}\int _{\mathbb {R}^2}[|\nabla v|^2+V(x) |f(v)|^2]dx- C\Vert v\Vert _X^{ 4-\mu }-C\Vert v\Vert _X^{2q}. \end{aligned}$$

Fixing $\Vert v\Vert _X=\rho <\frac{1}{4}$ and $\xi =\frac{4}{\rho }>1$ jointly with $f(t)t^{-1}$ is decreasing, one gets

$$\begin{aligned} \rho =\Vert v\Vert _X\le |\nabla v|_2+\frac{\rho }{4}+\frac{4}{\rho }\int _{\mathbb {R}^2}V(x)|f(v)|^2\,dx. \end{aligned}$$

Using the Young’s inequality, we arrive at

$$\begin{aligned} |\nabla v|_2 \le \frac{\rho }{4}+\frac{1}{\rho }|\nabla v|^{2}_2 \end{aligned}$$

from where it follows that

$$\begin{aligned} \frac{\rho }{4}\le \frac{1}{\rho }|\nabla v|^{2}_2+\frac{4}{\rho }\int _{\mathbb {R}^2}V(x)|f(v)|^2\,dx \end{aligned}$$

and so,

$$\begin{aligned} \mathcal {Q}(v) \triangleq \int _{\mathbb {R}^2}[|\nabla v|^2+V(x) |f(v)|^2]dx\ge \frac{\rho ^2}{16}. \end{aligned}$$

Hence,

$$\begin{aligned} \mathcal {J}_f(v)\ge \frac{\rho ^2}{16}- C\rho ^{ 4-\mu }-C\rho ^{2q}>0, \end{aligned}$$

there is a sufficiently small constant $\rho >0$ such that Point-(i) holds.

(ii) From $(h_3)$, we know that

$$\begin{aligned} \lim _{t \rightarrow +\infty }\frac{H(t)}{t^2}=+\infty . \end{aligned}$$

Setting $\varphi \in C_{0}^{\infty }(\mathbb {R}^2)$ with $\varphi (x) \ge 0$ for all $x \in \mathbb {R}^2$, a standard argument gives

$$\begin{aligned} \lim _{t \rightarrow +\infty }\mathcal {J}_f(t\varphi )=-\infty , \end{aligned}$$

then Point-(ii) occurs with $e=t_0\varphi$ and $t_0 \approx +\infty$. The proof is complete. $\square$

As a consequence of Proposition 2.2 and Lemma 2.4, we obtain the existence of (C) sequence of $\mathcal {J}_f$ at the level c, that is, $\mathcal {J}_f(v_n)\rightarrow c$ and $(1+\Vert v_n\Vert _X)\Vert \mathcal {J}_f^\prime (v_n)\Vert _{X^{-1}}\rightarrow 0$. To restore the compactness of $\{v_n\}$, we firstly derive the upper estimate for the mountain-pass level c. With this aim in mind, we shall deal with it by $(h_2)-(h_3)$. Motivated by [4, 7, 15, 19, 22, 27, 50], we consider the Moser sequence defined by

$$\begin{aligned} \overline{w}_n(x)\triangleq \frac{1}{\sqrt{2\pi }} \left\{ \begin{array}{ll} \sqrt{\log n}, &{}\quad \text {if}~0\le |x|\le \frac{1}{n}, \\ \frac{\log \left( \frac{1}{|x|}\right) }{\sqrt{\log n}}, &{} \quad \text {if}~ \frac{1}{n} <|x|\le 1, \\ 0, &{} \quad \text {if}~|x|>1. \end{array} \right. \end{aligned}$$

Lemma 2.5

Let (V) and (1.8) with $(h_1)-(h_4)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, then $0<\varrho \le c<c^*\triangleq \frac{\pi (4-2\beta -\mu )}{4\alpha _0}$.

Proof

Firstly, we obtain $c\ge \varrho >0$ by Lemma 2.4-(i). Due to Proposition 2.2, one could find that $c=\inf _{\gamma \in \Gamma }\max _{t\in (0,1]}\mathcal {J}_f(\gamma (t)) \le \inf _{v\in X\backslash \{0\}}\max _{t>0}\mathcal {J}_f(tv)$. So, we must show that there is a function $w\in X\backslash \{0\}$ such that $\max _{t>0}\mathcal {J}_f(tw)<c^*$. It follows from some elementary computations that

$$\begin{aligned} \Vert \overline{w}_n\Vert ^2_{H ^1(\mathbb {R}^2)}&\le 1+\delta _n, \end{aligned}$$

where

$$\begin{aligned} \delta _n\triangleq \frac{1}{4\log n}-\frac{1}{4n^2\log n}-\frac{1}{2n^2} >0. \end{aligned}$$

(2.5)

Set $w_n\triangleq \overline{w}_n/\sqrt{1+\delta _n}\in H ^1(\mathbb {R}^2)\backslash \{0\}$ and since $\textrm{supp}{\hspace{.05cm}}w_n\subset B_1(0)$, then $w_n\in X\backslash \{0\}$ and $\Vert w_n\Vert _{H ^1(\mathbb {R}^2)}\le 1$. We claim that there exists a $n\in \mathbb {N}^+$ such that

$$\begin{aligned} \max _{t>0}\mathcal {J}_f(tw_n)<c^*. \end{aligned}$$

(2.6)

Otherwise, for all $n\in \mathbb {N}^+$, there is a $t_n>0$ corresponding to the maximum point of $\max _{t>0}\mathcal {J}_f(tw_n)$

$$\begin{aligned} \langle \mathcal {J}_f^\prime (t_nw_n),t_nw_n\rangle =0~\text {and}~ \mathcal {J}_f(t_nw_n)= \max _{t>0}\mathcal {J}_f(tw_n)\ge c^* \end{aligned}$$

(2.7)

in which of the first formula together with $\Vert w_n\Vert _X\le 1$ implies that

$$\begin{aligned} t_n^2\ge \int _{\mathbb {R}^2}\Bigg (\int _{\mathbb {R}^2}\frac{H(f(t_nw_n(y)))}{|y|^\beta |x-y|^\mu }dy\Bigg ) \frac{h(f(t_nw_n))f^\prime (t_nw_n)t_nw_n}{|x|^\beta }dx. \end{aligned}$$

(2.8)

Since $H(s)\ge 0$ for all $s\in \mathbb {R}$, by $(f_2)$, we can infer from (2.7) and $\Vert w_n\Vert _{H ^1(\mathbb {R}^2)}\le 1$ that

$$\begin{aligned} t_n^2\ge 2c^*=\frac{(4-2\beta -\mu )\pi }{2\alpha _0},~\forall n\in \mathbb {N}^+. \end{aligned}$$

(2.9)

Combining $(h_2)$ and $(h_3)$, for all $\varepsilon \in (0,\beta _0)$, there is a constant $R_\varepsilon >0$ such that

$$\begin{aligned} H(s)h(s)s\ge M_0^{-1}(\beta _0-\varepsilon )s^{\vartheta +1}e^{2\alpha _0 s^4},~\forall s\ge R_\varepsilon \end{aligned}$$

which together with $(f_5)-(f_6)$ and (2.8)–(2.9) shows that

$$\begin{aligned} t_n^2&\ge \int _{B_{1/n}(0)} \Bigg (\int _{B_{1/n}(0)}\frac{ H(f(t_nw_n(y)))}{|y|^\beta |x-y|^\mu }dy\Bigg ) \frac{h(f(t_nw_n))f^\prime (t_nw_n)t_nw_n(x)}{|x|^\beta }dx \nonumber \\&\ge \frac{1}{2}\int _{B_{1/n}(0)} \Bigg (\int _{B_{1/n}(0)}\frac{ H(f(t_nw_n(y)))}{|y|^\beta |x-y|^\mu }dy\Bigg ) \frac{h(f(t_nw_n))f(t_nw_n)}{|x|^\beta }dx\nonumber \\&\ge \frac{1}{2}M_0^{-1}(\beta _0-\varepsilon )t_n^{\frac{\vartheta +1}{2}}\Bigg (\frac{\log n}{2\pi (1+\delta _n)}\Bigg )^{\frac{\vartheta +1}{4}} \Bigg (e^{ 2\alpha _0 t_n^2\pi ^{-1}(1+\delta _n)^{-1}\log n}\Bigg )\frac{n^{2\beta +\mu }}{2}|B_{1/n}(0)|^2\nonumber \\&=\frac{\pi ^2}{2M_0} (\beta _0-\varepsilon )t_n^{\frac{\vartheta +1}{2}}\Bigg (\frac{1}{2\pi (1+\delta _n)}\Bigg )^{\frac{\vartheta +1}{4}} e^{ [2\alpha _0 t_n^2\pi ^{-1}(1+\delta _n)^{-1}-(4-2\beta -\mu )]\log n+\frac{\vartheta +1}{2}\log (\log n)}. \end{aligned}$$

(2.10)

By (2.9) and $(\vartheta +1)\log (\log n)/2>0$, we can deduce that

$$\begin{aligned}{} & {} (1-\vartheta )\log t_n \ge \log \Bigg [\frac{\pi ^2}{2 M_0} (\beta _0-\varepsilon ) \Bigg (\frac{1}{2\pi (1+\delta _n)}\Bigg )^{\frac{\vartheta +1}{4}} \Bigg ] \nonumber \\{} & {} \quad +[2\alpha _0 t_n^2\pi ^{-1}(1+\delta _n)^{-1}-(4-2\beta -\mu )]\log n. \end{aligned}$$

(2.11)

If $\{t_n\}$ is unbounded, up to a subsequence if necessary, we can assume that $t_n\rightarrow +\infty$ and then

$$\begin{aligned}{} & {} \frac{(1-\vartheta )\log t_n}{t_n^2} \ge t_n^{-2}\log \Bigg [\frac{\pi ^2}{2 M_0} (\beta _0-\varepsilon ) \Bigg (\frac{1}{2\pi (1+\delta _n)}\Bigg )^{\frac{\vartheta +1}{4}} \Bigg ] \\{} & {} \quad +[2\alpha _0 \pi ^{-1}(1+\delta _n)^{-1}-t_n^{-2}(4-2\beta -\mu )]\log n \end{aligned}$$

which together with $\delta _n\rightarrow 0$ in (2.5) yields a contradiction if we tend $n\rightarrow \infty$. Thereby, passing to a subsequence if necessary, there exists a positive constant $t_0$ such that

$$\begin{aligned} \lim _{n\rightarrow \infty } t_n^2=t_0^2\ge \frac{(4-2\beta -\mu )\pi }{2\alpha _0}, \end{aligned}$$

where (2.9) gives the inequality. Moreover, we conclude that $t_0^2= (4-2\beta -\mu )\pi /2\alpha _0$. Otherwise, we obtain a contradiction by letting $n\rightarrow \infty$ in (2.11). Let’s tend $n\rightarrow \infty$ in (2.10), there holds

$$\begin{aligned} \frac{(4-2\beta -\mu )\pi }{2\alpha _0} =t_0^2\ge \frac{\pi ^2}{2 M_0} (\beta _0-\varepsilon )t_0^{\frac{\vartheta +1}{2}}\Bigg (\frac{1}{2\pi }\Bigg )^{\frac{\vartheta +1}{4}} \lim _{n\rightarrow \infty } e^{ \frac{\vartheta +1}{4}\log (\log n)}=+\infty , \end{aligned}$$

a contradiction. So, (2.6) holds true. The proof is complete. $\square$

Next, we mainly try to conclude that each $(C)_c$ sequence of $\mathcal {J}_f$ is uniformly bounded. To this aim, we shall split it two cases. In other words, we prefer to divide to $\delta \in (\frac{1}{2},1)$ and $\delta =\frac{1}{2}$, respectively. Let us firstly handle the case $\delta \in (\frac{1}{2},1)$ as follows.

Lemma 2.6

Let (V) and (1.8) with $(h_1)-(h_4)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$. If $\delta \in (\frac{1}{2},1)$ in $(h_4)$, then every $(C)_c$ sequence $\{v_n\}\subset X$ of $\mathcal {J}_f$ is uniformly bounded.

Proof

Let $\{v_n\}\subset X$ be a $(C)_c$ sequence of $\mathcal {J}_f$, that is, $\mathcal {J}_f(v_n)\rightarrow c$ and $(1+\Vert v_n\Vert _X)\Vert \mathcal {J}^\prime _f(v_n)\Vert _{X^{-1}}\rightarrow 0$, then one immediately obtains

$$\begin{aligned}{} & {} \int _{\mathbb {R}^2}[|\nabla v_n|^2+V(x)|f(v_n)|^2]dx\nonumber \\{} & {} \qquad - \int _{\mathbb {R}^2}[|x|^{-\mu }*(|x|^{-\beta }H(f(v_n)))] |x|^{-\beta }H(f(v_n))dx= 2c+o_n(1) \nonumber \\ \end{aligned}$$

(2.12)

and for all $\{\psi _n\}\subset X$, there holds

$$\begin{aligned} \begin{aligned}&\Bigg | \int _{\mathbb {R}^2}[\nabla v_n \nabla \psi _n+V(x) f(v_n)f^\prime (v_n)\psi _n]dx\\&\qquad -\int _{\mathbb {R}^2}[|x|^{-\mu }*(|x|^{-\beta }H(f(v_n)))] |x|^{-\beta }h(f(v_n))f^\prime (v_n)\psi _ndx\Bigg |\\&\quad \le \Vert \mathcal {J}^\prime _f(v_n)\Vert _{X^{-1}}\Vert \psi _n\Vert _X, \end{aligned}\nonumber \\ \end{aligned}$$

(2.13)

where $o_n(1)\rightarrow 0$ as $n\rightarrow \infty$. Without loss of generality, we shall suppose that $v_n\ne 0$. Furthermore, we can suppose $v_n>0$ by $(h_1)$. Motivated by [53], we let $\psi _n= {H(f(v_n))}/[{h(f(v_n))f^\prime (v_n)}]$. Since $\{v_n\}\subset X$, by using (1.12) and $(f_6)$, one has

$$\begin{aligned} \int _{\mathbb {R}^2}V(x)\psi _n^2dx\le 4\int _{\mathbb {R}^2}V(x)v_n^2dx<+\infty \end{aligned}$$

and the computation

$$\begin{aligned} \nabla \psi _n=\frac{h^2(f(v_n))-H(f(v_n))h^\prime (f(v_n))+H(f(v_n))h(f(v_n))f(v_n)[f^\prime (v_n)]^2}{h^2(f(v_n))}\nabla v_n \end{aligned}$$

which together with $(h_4)$, (1.7) and (1.13) gives that

$$\begin{aligned} \int _{\mathbb {R}^2}|\nabla \psi _n|^2dx\le 4(1-\delta )^2\int _{\mathbb {R}^2}|\nabla v_n|^2dx <+\infty . \end{aligned}$$

Therefore, $\{\psi _n\}\subset X$ and it could be applied in (2.13). Moreover, by means of $(h_4)$, (1.7) and (1.13) again,

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^2}[\nabla v_n \nabla \psi _n+V(x) f(v_n)f^\prime (v_n)\psi _n]dx =2(1-\delta )\int _{\mathbb {R}^2} |\nabla v_n|^2dx\\&\qquad + \int _{\mathbb {R}^2}V(x)\frac{H(f(v_n))f(v_n)}{h(f(v_n))}dx \\&\quad \le 2(1-\delta )\int _{\mathbb {R}^2}[|\nabla v_n|^2+V(x)|f(v_n)|^2]dx=2(1-\delta )\mathcal {Q}(v_n) \end{aligned} \end{aligned}$$

which jointly with (2.12) and (2.13) indicates that

$$\begin{aligned} \mathcal {Q}(v_n)&\le 2c+o_n(1)+ \int _{\mathbb {R}^2}[\nabla v_n \nabla \psi _n+V(x) f(v_n)f^\prime (v_n)\psi _n]dx +o_n(1)\Vert \psi _n\Vert _X \\&\le 2c+o_n(1)+2(1-\delta )\mathcal {Q}(v_n)+2 \Vert \mathcal {J}^\prime _f(v_n)\Vert _{X^{-1}} \Vert v_n\Vert _X, \end{aligned}$$

where we have exploted the fact that $\Vert \psi _n\Vert _X\le 2\Vert v_n\Vert _X$. By $\delta \in (\frac{1}{2},1)$, we can conclude that $\{\mathcal {Q}(v_n)\}$ is bounded. Since

$$\begin{aligned} \Vert v_n\Vert _X \le 1+ \mathcal {Q}(v_n), \quad \forall n \in \mathbb {N}, \end{aligned}$$

we can accomplish the proof of this lemma. $\square$

Conversely, the case $\delta =\frac{1}{2}$ is totally distinct, and so it is necessary to take a more delicate analysis. In light of this, we need to derive the following result.

Lemma 2.7

Let (V) and (1.8) with $(h_1)-(h_4)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$. If $\{v_n\}\subset E$ satisfies $v_n\rightharpoonup 0$ in E, $v_n\rightarrow 0$ a.e. in $\mathbb {R}^2$ and $|\nabla v_n|^2_2<\frac{\pi (4-2\beta -\mu )}{2\alpha _0}$, then

$$\begin{aligned} \displaystyle \lim _{n\rightarrow \infty }\int _{\mathbb {R}^2}\big [|x|^{-\mu }*\big (|x|^{-\beta }H(f({v}_n))\big )\big ]|x|^{-\beta }H(f({v}_n))dx=0. \end{aligned}$$

Proof

Since $|\nabla v_n|_{2}^2<\frac{\pi (4-2\beta -\mu )}{2\alpha _0}$ and $X \hookrightarrow H^1(\mathbb {R}^2)$, it follows that $\Vert v_n\Vert _{H^1(\mathbb {R}^2)}$ is bounded. Thereby, we can argue as (2.4) to obtain

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(f(v_n(x)))H(f(v_n(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx \le C_\mu \Bigg (\int _{\mathbb {R}^2}\frac{|H(f(v_n))|^{\frac{4}{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{2}}\\&\quad \le C_\mu \Bigg (\int _{\mathbb {R}^2}\frac{ |v_n|^{2}}{|x|^{\frac{4\beta }{4-\mu }}}dx \Bigg )^{\frac{4-\mu }{2}} +C \Bigg (\int _{\mathbb {R}^2} \frac{|v_n|^{\frac{4q}{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}} \left( e^{\frac{8\alpha }{4-\mu }|v|^2}-1\right) dx\Bigg )^{\frac{4-\mu }{2}}\\&\quad \le C_\mu \Bigg (\int _{\mathbb {R}^2}\frac{ |v_n|^{2}}{|x|^{\frac{4\beta }{4-\mu }}}dx \Bigg )^{\frac{4-\mu }{2}} +C \Bigg (\int _{\mathbb {R}^2}\frac{|v_n|^{\frac{4q\nu }{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}} dx\Bigg )^{\frac{4-\mu }{2\nu }}\\&\qquad \times \left( \int _{\mathbb {R}^2} \frac{\left( e^{\frac{8\alpha \nu ^\prime (1+\epsilon )^2|\nabla v_n|_{2}^2}{4-\mu }\Bigg (\frac{v_n}{(1+\epsilon )|\nabla v_n|_{2}} \Bigg )^2}-1\right) }{|x|^{\frac{4\beta }{4-\mu }}} dx\right) ^{\frac{4-\mu }{2\nu ^\prime }}\\&\quad \le C_\mu \Bigg (\int _{\mathbb {R}^2}\frac{ |v_n|^{2}}{|x|^{\frac{4\beta }{4-\mu }}}dx \Bigg )^{\frac{4-\mu }{2}}+C \Bigg (\int _{\mathbb {R}^2} \frac{|v_n|^{\frac{4q}{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}} dx\Bigg )^{\frac{4-\mu }{2\nu }} \end{aligned} \end{aligned}$$

yielding the desired result by Lemma 2.3 for $\epsilon \approx 0^+$. The proof is complete. $\square$

Lemma 2.8

Let (V) and (1.8) with $(h_1)-(h_4)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$. If $\delta =\frac{1}{2}$ in $(h_4)$, then every $(C)_c$ sequence $\{v_n\}\subset X$ of $\mathcal {J}_f$ is uniformly bounded whenever $c<c^*\triangleq \frac{\pi (4-2\beta -\mu )}{4\alpha _0}$.

Proof

We argue it by the contradiction and assume, up to a subsequence if necessary, that $\Vert v_n\Vert _X\rightarrow \infty$, then $v_n\in E$ for all fixed $n\in \mathbb {N}^+$ and $\Vert v_n\Vert \rightarrow \infty$. Define $\bar{v}_n=\sigma v_n/\Vert v_n\Vert$ with $\sigma = \sqrt{c+c^* }$, since $c<c^*$, we have that

$$\begin{aligned} 2c<\Vert \bar{v}_n\Vert ^2=\sigma ^2=c+c^*<2c^*. \end{aligned}$$

(2.14)

Going to a subsequence if necessary, there exists a function $\bar{v}\in E$ such that $\bar{v}_n\rightharpoonup \bar{v}$ in E. We claim that $\bar{v}\ne 0$ in $\mathbb {R}^2$. Otherwise, we suppose that $\bar{v}\equiv 0$ a.e. in $\mathbb {R}^2$. Combining (2.14) and Lemma 2.7,

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathcal {J}_f( \bar{v}_n)=\frac{1}{2} \lim _{n\rightarrow \infty } \int _{\mathbb {R}^2}[|\nabla \bar{v}_n|^2+V(x)|\bar{v}_n|^2]dx =\frac{\sigma ^2}{2}. \end{aligned}$$

(2.15)

Define $\varphi _n \triangleq f( v_n)/f^\prime ( v_n)$ and $\bar{\varphi }_n \triangleq f( \bar{v}_n)/f^\prime ( \bar{v}_n)$, then $|\varphi _n|\le 2|v_n|$ and $|\bar{\varphi }_n|\le 2|\bar{v}_n|$ by $(f_6)$ and

$$\begin{aligned} \nabla \varphi _n = \Bigg [1+\frac{f^2( v_n)}{1+f^2( v_n)}\Bigg ]\nabla v_n~\text {and}~ \nabla \bar{\varphi }_n = \Bigg [1+\frac{f^2( \bar{v}_n)}{1+f^2( \bar{v}_n)}\Bigg ]\nabla \bar{v}_n. \end{aligned}$$

So, $\Vert \varphi _n \Vert \le 2\Vert v_n\Vert$ and $\Vert \overline{\varphi }_n \Vert \le 2\Vert \overline{v}_n\Vert$, then $\varphi _n, \overline{\varphi }_n\in E$. It is simple to compute that $\max _{t\in (0,1]}\mathcal {J}_f(t\bar{v}_n)$ can be achieved at some $t_n\in (0,1)$ and then $\mathcal {J}^\prime _f(t_n\overline{v}_n)[t_n\bar{v}_n]=0$. Going to a subsequence if necessary, we claim that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\mathcal {J}^\prime _f(t_n\bar{v}_n)[f( t_n\bar{v}_n)/f^\prime ( t_n\bar{v}_n)]\le 0. \end{aligned}$$

(2.16)

In fact, we know that $f( t_n\bar{v}_n)/f^\prime ( t_n\bar{v}_n)\in E$ since $t_n\in (0,1)$ which makes the left side of (2.16) sense. Setting $\tilde{v}_n\triangleq t_n \bar{v}_n$, then, selecting a subsequence still denoted by itself, one has $\tilde{v}_n\rightharpoonup 0$ in E and $\Vert \tilde{v}_n\Vert ^2\le \Vert \bar{v}_n\Vert ^2=\sigma ^2$. Hence, similar to the proof of Lemma 2.7 and by $(f_6)$ in Lemma 2.1

$$\begin{aligned}{} & {} \limsup _{n\rightarrow \infty }\mathcal {J}^\prime _f(\tilde{v}_n)[f(\tilde{v}_n)/f^\prime (\tilde{v}_n)] = \limsup _{n\rightarrow \infty }\big (\mathcal {J}^\prime _f(\tilde{v}_n)[f(\tilde{v}_n)/f^\prime (\tilde{v}_n)] -2 \mathcal {J}^\prime _f(\tilde{v}_n)[ \tilde{v}_n ]\big ) \\{} & {} \quad = \limsup _{n\rightarrow \infty }\Bigg \{ -\int _{\mathbb {R}^2}|\nabla f(\tilde{v}_n)|^2dx-\int _{\mathbb {R}^2}V(x)[2f(\tilde{v}_n)f^\prime (\tilde{v}_n)\tilde{v}_n-|f(\tilde{v}_n)|^2]dx\\{} & {} \qquad + \int _{\mathbb {R}^2}\big [|x|^{-\mu }*\big (|x|^{-\beta }H(f(\tilde{v}_n))\big )\big ]|x|^{-\beta } h(f(\tilde{v}_n))[f(\tilde{v}_n)-2f^\prime (\tilde{v}_n )\tilde{v}_n]dx\Bigg \}\\{} & {} \quad =\limsup _{n\rightarrow \infty }\Bigg \{ -\int _{\mathbb {R}^2}|\nabla f(\tilde{v}_n)|^2dx-\int _{\mathbb {R}^2}V(x)[2f(\tilde{v}_n)f^\prime (\tilde{v}_n)\tilde{v}_n-|f(\tilde{v}_n)|^2]dx\Bigg )\le 0 \end{aligned}$$

showing the desired result. Since $\Vert v_n\Vert \rightarrow +\infty$ as $n\rightarrow \infty$, $\sigma t_n/\Vert v_n\Vert \in (0,1)$ for all sufficiently large $n\in \mathbb {N}$. Combining (1.14), (2.16) and $(f_9)$ in Lemma 2.1, we obtain

$$\begin{aligned} \mathcal {J}_f(\bar{v}_n)&\le \max _{t\in (0,1]}\mathcal {J}_f(t\bar{v}_n)= \mathcal {J}_f(t_n\overline{v}_n) \le \mathcal {J}_f(t_n\bar{v}_n)\nonumber \\&\quad -\frac{1}{4} \mathcal {J}_f^\prime (t_n\bar{v}_n)[f(t_n\bar{v}_n)/f^\prime (t_n\bar{v}_n)] +o_n(1)\nonumber \\&=\frac{1}{4}\int _{\mathbb {R}^2}\big [|\nabla f(t_n\bar{v}_n)|^2+V(x)|f(t_n\bar{v}_n)|^2\big ]dx\nonumber \\&\quad + \frac{1}{4}\int _{\mathbb {R}^2}[|x|^{-\mu }*(|x|^{-\beta }H(f(t_n\bar{v}_n)))]|x|^{-\beta }[h(f(t_n\bar{v}_n))f(t_n\bar{v}_n)\nonumber \\&\quad -2H(f(t_n\bar{v}_n))]dx+o_n(1)\nonumber \\&\le \frac{1}{4}\int _{\mathbb {R}^2}\big [|\nabla f(v_n)|^2+V(x)|f(v_n)|^2\big ]dx\nonumber \\&\quad + \frac{1}{4}\int _{\mathbb {R}^2}[|x|^{-\mu }*(|x|^{-\beta }H(f(v_n)))]|x|^{-\beta }[h(f(v_n))f(v_n)\nonumber \\&\quad -2H(f(v_n))]dx+o_n(1)\nonumber \\&\quad =\mathcal {J}_f(v_n)-\frac{1}{4} \mathcal {J}_f^\prime (v_n)[f(v_n)/f^\prime (v_n)]+o_n(1)= \mathcal {J}_f(v_n)+o_n(1). \end{aligned}$$

(2.17)

Let’s recall that $\{v_n\}$ is a $(C)_c$ sequence of J, taking the limit $n\rightarrow \infty$ in (2.17), we would conclude that $\sigma ^2\le 2c$ by (2.15), a contradiction to (2.14). Consequently, $\bar{v}\ne 0$ in $\mathbb {R}^2$ and then there exists a constant $R>0$ such that $B_R(0)\cap \mathcal {B}$ admits positive Lebesgue measure, where $\mathcal {B} \triangleq \{x\in \mathbb {R}^2|\bar{v}(x)\ne 0\}$. Since $\Vert v_n\Vert \rightarrow \infty$, one knows $|v_n|\rightarrow \infty$ on $B_R(0)\cap \mathcal {B}$. It infers from $(h_3)$ that $H(s)/|s|^2\rightarrow +\infty$ as $|s|\rightarrow \infty$. Due to $(f_5)$ in Lemma 2.1 and the Fatou’s lemma, we have

$$\begin{aligned} \begin{aligned}&\frac{1}{\Vert v_n\Vert ^2}\int _{\mathbb {R}^2}[|x|^{-\mu }*(|x|^{-\beta }H(f(v_n)))]|x|^{-\beta }H(f(v_n))dx\\&\quad = \int _{\mathbb {R}^2}\Bigg (\int _{\mathbb {R}^2}\frac{ H(f(v_n))}{|y|^{ \beta }|x-y|^\mu \Vert v_n\Vert }dy\Bigg ) \frac{ H(f(v_n))}{|x|^{\beta }\Vert v_n\Vert }dx \\&\quad \ge \frac{1}{\sigma ^2R^{\mu +2\beta }}\Bigg (\int _{B_R(0)\cap \mathcal {B}} \frac{ H(f(v_n))}{ f^2(v_n) } \frac{f^2(v_n)}{|v_n|} |\bar{v}_n|dx\Bigg )^2 \rightarrow +\infty ~\text {as}~n\rightarrow \infty . \end{aligned} \end{aligned}$$

Recalling that $\{v_n\}$ is a $(C)_c$ sequence of $\mathcal {J}_f$, then

$$\begin{aligned} 0= & {} \liminf _{n\rightarrow \infty }\frac{\mathcal {J}_f(v_n)}{\Vert v_n\Vert ^2}\\\le & {} \frac{1}{2}-\limsup _{n\rightarrow \infty } \frac{1}{\Vert v_n\Vert ^2}\int _{\mathbb {R}^2} [|x|^{-\mu }*(|x|^{-\beta }H(f(v_n)))]|x|^{-\beta }H(f(v_n)) dx=-\infty , \end{aligned}$$

a contradiction. In summary, we finish the proof of this lemma. $\square$

Following [9, Lemma 4.6], we have to establish the following result before showing that any weak limit of the $(C)_c$ sequence $\{v_n\}\subset X$ of $\mathcal {J}_f$ can be adopted to construct the existence of nontrivial solutions to Eq. (1.1). More precisely, we introduce the following compactness result.

Lemma 2.9

If $\{v_n\}\subset H^1(\mathbb {R}^2)$ satisfies $v_n\rightharpoonup v_0$ in $H^1(\mathbb {R}^2)$ as $n\rightarrow \infty$ and there exists a constant $K_0>0$ such that

$$\begin{aligned} \sup _{n\in \mathbb {N}}\int _{\mathbb {R}^2} \Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_n))}{|y|^\beta |x-y|^\mu }dy\Bigg )\frac{h(f(v_n))f^\prime (v_n)v_n}{|x|^\beta }dx\le K_0. \end{aligned}$$

(2.18)

Then, going to a subsequence if necessary, there holds

$$\begin{aligned}{} & {} \lim _{n\rightarrow \infty }\int _{\mathbb {R}^2} \Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_n))}{|y|^\beta |x-y|^\mu }dy\Bigg )\frac{H(f(v_n))}{|x|^\beta }dx\nonumber \\{} & {} \quad = \int _{\mathbb {R}^2} \Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_0))}{|y|^\beta |x-y|^\mu }dy\Bigg )\frac{H(f(v_0))}{|x|^\beta }dx. \end{aligned}$$

(2.19)

Moreover, for all $\psi \in C_{0}^\infty (\mathbb {R}^2)$, going to a subsequence if necessary, we can conclude that

$$\begin{aligned}{} & {} \lim _{n\rightarrow \infty }\int _{\mathbb {R}^2} \Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_n))}{|y|^\beta |x-y|^\mu }dy\Bigg )\frac{h(f(v_n))f^\prime (v_n)\psi }{|x|^\beta }dx\nonumber \\{} & {} \quad = \int _{\mathbb {R}^2} \Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_0))}{|y|^\beta |x-y|^\mu }dy\Bigg )\frac{h(f(v_n))f^\prime (v_n)\psi }{|x|^\beta }dx. \end{aligned}$$

(2.20)

We must stress here that the ideas in the proof Lemma 2.9 can date back to [9, Lemma 4.6], but there exist some obvious differences because of the quasilinear term and the non-radial symmetrically work space $H^1(\mathbb {R}^2)$.

Proof

Combining (2.18) and the Fatou lemma, one has

$$\begin{aligned} \int _{\mathbb {R}^2} \Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_0))}{|y|^\beta |x-y|^\mu }dy\Bigg )\frac{h(f(v_0))f^\prime (v_0)v_0}{|x|^\beta }dx\le K_0. \end{aligned}$$

(2.21)

In view of $(h_2)$, one has that

$$\begin{aligned} 0\le \lim _{s\rightarrow +\infty }\frac{H(s)}{h(s)s}\le \lim _{s\rightarrow +\infty }\frac{M_0}{s^{\vartheta +1}}=0 \end{aligned}$$

and for all $\varepsilon >0$, there exists a constant $\overline{s}=\overline{s}(\varepsilon )>1$ such that

$$\begin{aligned} H(s)\le \varepsilon h(s)s,~\forall s\ge \overline{s} \end{aligned}$$

which together with $(f_6)$ in Lemma 2.1, (2.18) and (2.21) gives that

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \sup _{n\in \mathbb {N}}\int _{\{f(v_n)\ge \overline{s}\}} \Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_n(y)))}{|y|^\beta |x-y|^\mu }dy\Bigg )\frac{H(f(v_n(x)))}{|x|^\beta }dx \le 2K_0\varepsilon ,\\ \displaystyle \int _{\{f(v_0)\ge \overline{s}\}}\Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_0(y)))}{|y|^\beta |x-y|^\mu }dy\Bigg )\frac{H(f(v_0(x)))}{|x|^\beta }dx\le 2K_0\varepsilon . \end{array} \right. \end{aligned}$$

(2.22)

Arguing as in [9, Appendix], we can derive that

$$\begin{aligned}{} & {} \limsup _{n\rightarrow \infty }\int _{\{f(v_0)=\overline{s}\} \cap \{f(v_n)< \overline{s}\}} \Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_n(y)))}{|y|^\beta |x-y|^\mu }dy\Bigg )\frac{H(f(v_n(x)))}{|x|^\beta }dx\nonumber \\{} & {} \quad \le \overline{C}\varepsilon , \end{aligned}$$

(2.23)

where $\overline{C}>0$ is independent of $n\in \mathbb {N}$. Let’s define

$$\begin{aligned} \Theta _n&\triangleq \int _{\{f(v_0)\not =\overline{s}\}\cap \{f(v_n)< \overline{s}\}} \Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_n(y)))}{|y|^\beta |x-y|^\mu }dy \Bigg )\frac{H(f(v_n(x)))}{|x|^\beta }dx\\&=\int _{\mathbb {R}^2}\Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_n(x)))\chi _{\{\{f(v_0)\not =\overline{s}\}\cap \{f(v_n)< \overline{s}\}\}}(y)}{|y|^\beta |x-y|^\mu }dy\Bigg )\frac{H(f(v_n(x)))}{|x|^\beta }dx\\&=\int _{\mathbb {R}^2} \frac{\xi _nH(f(v_n))}{|x|^\beta } dx \end{aligned}$$

and similarly

$$\begin{aligned} \Theta _0 \triangleq \int _{\{f(v_0)< \overline{s}\}} \Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_0(y)))}{|y|^\beta |x-y|^\mu }dy\Bigg )\frac{H(f(v_0(x)))}{|x|^\beta }dx =\int _{\mathbb {R}^2} \frac{\xi _0 H(f(v_0))}{|x|^\beta } dx, \end{aligned}$$

where

$$\begin{aligned} \left\{ \begin{array}{ll} \xi _n(x)\triangleq \displaystyle \int _{\mathbb {R}^2}\frac{H(f(v_n(y)))\chi _{\{\{f(v_n)\not =\overline{s}\}\cap \{f(v_n)< \overline{s}\}\}}(y)}{|y|^\beta |x-y|^{\mu }}dy,~\quad x\in \mathbb {R}^2, \\ \xi _0(x)\triangleq \displaystyle \int _{\mathbb {R}^2}\frac{H(f(v_0(y)))\chi _{\{f(v_0)<\overline{s}\}}(y)}{|y|^\beta |x-y|^{\mu }}dy,~\quad x\in \mathbb {R}^2. \end{array} \right. \end{aligned}$$

Claim 1. $\{\xi _n\}$ is uniformly bounded in $n\in \mathbb {N}$ and $\xi _n\rightarrow \xi _0$ a.e. in $\mathbb {R}^2$.

Verification: Let $\varepsilon =1$ and $q=1$ in (2.3), then one easily sees that there is a constant $C(\overline{s})>0$ such that

$$\begin{aligned} |H(s)|\le C(\overline{s})|s|^{(4 -\mu )/2},~\forall |s|\le \overline{s} \end{aligned}$$

which together with $(f_2)$ in Lemma 2.1 implies that

$$\begin{aligned} |\xi _n(x)| \le C(\overline{s})\Bigg ( \int _{ |x-y|< 1 }\frac{ | v_n(y)|^{\frac{ 4 -\mu }{2}}}{|y|^\beta |x-y|^{\mu }}dy +\int _{ |x-y|\ge 1 }\frac{ | v_n(y)|^{\frac{ 4 -\mu }{2}}}{|y|^\beta |x-y|^{\mu }}dy \Bigg ). \end{aligned}$$

(2.24)

According to the Holder’s inequality, we can obtain the following two formulas

$$\begin{aligned}{} & {} \int _{ |x-y|< 1 }\frac{ | v_n(y)|^{\frac{ 4 -\mu }{2}}}{|y|^\beta |x-y|^{\mu }}dy\nonumber \\{} & {} \quad \le \Bigg (\int _{\mathbb {R}^2}\frac{|v_n|^{\frac{2(4-\mu )}{2+\beta -\mu }}}{|y|^{\frac{4\beta }{2+\beta -\mu }}} dy \Bigg )^{\frac{2+\beta -\mu }{4}} \Bigg (\int _{|x-y|< 1} \frac{1}{|x-y|^{\frac{4\mu }{2-\beta +\mu }}}dy \Bigg )^{\frac{2-\beta +\mu }{4}} \end{aligned}$$

(2.25)

and

$$\begin{aligned} \int _{ |x-y|\ge 1}\frac{ | v_n(y)|^{\frac{ 4 -\mu }{2}}}{|y|^\beta |x-y|^{\mu }}dy\le \Bigg (\int _{\mathbb {R}^2}\frac{ |v_n|^{2} }{ |y|^{\frac{4\beta }{4-\mu }} }dy\Bigg )^{\frac{4-\mu }{4}} \Bigg (\int _{|x-y|\ge 1}\frac{1}{|x-y|^4}dy\Bigg )^{\frac{\mu }{4}}. \nonumber \\ \end{aligned}$$

(2.26)

Since

$$\begin{aligned} \max \Bigg \{ \frac{4\beta }{2+\beta -\mu },\frac{4\mu }{2-\beta +\mu },\frac{4\beta }{4-\mu } \Bigg \}<2<\frac{2(4-\mu )}{2+\beta -\mu }. \end{aligned}$$

Combining (2.24), (2.25), (2.26) and Lemma 2.3, we deduce that $\{\xi _n\}$ is uniformly bounded. Thereby, with (2.25) and (2.26) in hand, we can follow [9, Appendix] to show that $\xi _n\rightarrow \xi _0$ a.e. in $\mathbb {R}^2$.

Claim 2. $\Theta _n\rightarrow \Theta _0$ as $n\rightarrow \infty$.

Verification: Thanks to the HLS inequality and $(f_2)$ in Lemma 2.1,

$$\begin{aligned}{} & {} |x|^{-\beta }\xi _nH(f(v_n))\chi _{\{|u_n|<\overline{s}\}}\nonumber \\{} & {} \quad \le C \left[ |x|^{-\mu }*\left( |x|^{-\beta }|v_n|^{\frac{4-\mu }{2}}\right) \right] |x|^{-\beta }|v_n|^{\frac{4-\mu }{2}} \in L^1(\mathbb {R}^2) \end{aligned}$$

and $[|x|^{-\mu }*(|x|^{-\beta }|v_n|^{\frac{4-\mu }{2}})] |x|^{-\beta }|v_n|^{\frac{4-\mu }{2}}\rightarrow [|x|^{-\mu }*(|x|^{-\beta }|v_0|^{\frac{4-\mu }{2}})] |x|^{-\beta }|v_0|^{\frac{4-\mu }{2}}$ in $L^1(\mathbb {R}^2)$ by Lemma 2.3, we apply the generalized Dominated Convergence theorem together with the Claim 1 to obtain

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{\mathbb {R}^2}|x|^{-\beta }\xi _nH(f(v_n))\chi _{\{f(v_n)<\overline{s}\}}dx =\int _{\mathbb {R}^2}|x|^{-\beta }\xi _0H(f(v_0))\chi _{\{f(v_0)<\overline{s}\}}dx. \end{aligned}$$

which together with (2.22) yields that

$$\begin{aligned} \lim _{n\rightarrow \infty }|\Theta _n-\Theta _0|&=\lim _{n\rightarrow \infty }\Bigg |\int _{\{f(v_n)\ge \overline{s}\}} |x|^{-\beta }\xi _n H(f(v_n))dx\\&\quad + \int _{\{f(v_n)< \overline{s}\}} |x|^{-\beta }\xi _n H(f(v_n))dx\\&\quad -\int _{\{f(v_0)\ge \overline{s}\}} |x|^{-\beta }\xi _0 H(f(v_0))dx\\&\quad - \int _{\{f(v_0)< \overline{s}\}} |x|^{-\beta }\xi _0 H(f(v_0))dx\Bigg |\\&\le 4K_0\varepsilon +\lim _{n\rightarrow \infty }\Bigg |\int _{\{f(v_n)< \overline{s}\}} |x|^{-\beta }\xi _n H(f(v_n))dx\\&\quad -\int _{\{f(v_0)< \overline{s}\}} |x|^{-\beta }\xi _0 H(f(v_0))dx\Bigg |\\&=4K_0\varepsilon +\lim _{n\rightarrow \infty }\Bigg |\int _{\mathbb {R}^2} |x|^{-\beta }\xi _n H(f(v_n))\chi _{\{f(v_n)< \overline{s}\}}dx\\&\quad -\int _{\mathbb {R}^2} |x|^{-\beta }\xi _0 H(f(v_0))\chi _{\{f(v_0)< \overline{s}\}}dx\Bigg |\\&=4K_0\varepsilon \end{aligned}$$

showing the Claim 2 since $\varepsilon >0$ is arbitrary. In consideration of the given $\varepsilon >0$, there is a sufficiently large $n_0\in \mathbb {N}^+$ such that $|\Theta _n-\Theta _0|\le \varepsilon$ for all $n\ge n_0$. Now, as a consequence of (2.22)–(2.23), we can conclude that

$$\begin{aligned}{} & {} \Bigg |\int _{\mathbb {R}^2}[|x|^{-\mu }*(|x|^{-\beta }H(f(v_n)))] |x|^{-\beta }H(f(v_n))dx\\{} & {} \qquad - \int _{\mathbb {R}^2}[|x|^{-\mu }*(|x|^{-\beta }H(f(v_0)))] |x|^{-\beta }H(f(v_0)) dx\Bigg |\\{} & {} \quad \le \int _{\{f(v_n)\ge \overline{s}\}}[|x|^{-\mu }*(|x|^{-\beta }H(f(v_n)))]|x|^{-\beta }H(f(v_n))dx \\{} & {} \qquad +\int _{\{f(v_0)\ge \overline{s}\}}[|x|^{-\mu }*(|x|^{-\beta }H(f(v_0)))] |x|^{-\beta }H(f(v_0)) dx\\{} & {} \qquad +\int _{\{|u_0|=\overline{s}\} \cap \{f(v_n)< \overline{s}\}} [|x|^{-\mu }*(|x|^{-\beta }H(f(v_n)))]|x|^{-\beta }H(f(v_n))dx+|\Theta _n-\Theta _0|\\{} & {} \quad \le (4K_0+\overline{C}+1)\varepsilon , ~\forall n\ge n_0. \end{aligned}$$

Thus, (2.19) holds true. Based on the above calculations, the proof of (2.20) is trivial and we could omit it. The proof is complete. $\square$

Now, it can be conclude that the variational functional $\mathcal {J}_f$ admits a nontrivial critical point. In other words, we success in finding a nontrivial solutions for Eq. (1.1). Thus, we can establish the following existence result.

Lemma 2.10

Let (V) and (1.8) with $(h_1)-(h_4)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, then Eq. (1.1) has at least a nontrivial solution in X.

Proof

By means of Proposition 2.2, Lemmas 2.4 and 2.5, we know that $\mathcal {J}_f$ admits a $(C)_c$ sequence, saying it $\{v_n\}$, in X at the level $0<c<c^*$. By Lemmas 2.6 and 2.8, one sees that $\{v_n\}$ is uniformly bounded in X and then (2.18) holds true. Recalling Lemma 2.3, going to a subsequence if necessary, there is a function $v_0\in X$ such that $v_n\rightharpoonup v_0$ in X, $v_n\rightarrow v_0$ in $L^q(\mathbb {R}^2,|x|^{-s}dx)$ for all $s\in (0,2)$ and $q\in [2,+\infty )$, and $v_n\rightarrow v_0$ a.e. in $\mathbb {R}^2$. According to (2.19), we have that $\mathcal {J}_f^\prime (v_0)=0$ which indicates that $u_0=f(v_0)$ is a solution of Eq. (1.1). By $(f_1)$ in Lemma 2.1, the remaining part is to verify that $v_0\ne 0$. Arguing it indirectly, we suppose that $v_0\equiv 0$. By using (2.19) and Lemma 2.5, there holds

$$\begin{aligned}{} & {} 0< \limsup _{n\rightarrow \infty }\mathcal {Q}(v_n)\nonumber \\{} & {} \quad = \limsup _{n\rightarrow \infty }\int _{\mathbb {R}^2} \Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_n))}{|y|^\beta |x-y|^\mu }dy\Bigg )\frac{H(f(v_n))}{|x|^\beta }dx +2c=2c<2c^*.\nonumber \\ \end{aligned}$$

(2.27)

Thereby, we shall chose $\alpha >\alpha _0$ sufficiently close to $\alpha _0$ and $\nu ^\prime >1$ sufficiently close to 1 in such a way that $1/v+1/v^\prime =1$ and

$$\begin{aligned} |\nabla v_n |^2 _2< \frac{\pi (4-2\beta -\mu )}{2\alpha _0} (1-\epsilon ) ~\text {for some suitable}~\epsilon \in (0,1). \end{aligned}$$

(2.28)

With this choice of $\alpha >\alpha _0$ and $\nu >1$, by (1.3), (2.3) and (2.28), we apply (1.11) to derive

$$\begin{aligned}{} & {} \int _{\mathbb {R}^2}\Bigg [\frac{1}{|x|^{\mu }}*\frac{h(f(v_n))f^\prime (v_n)v_n}{|x|^\beta }\Bigg ]\frac{h(f(v_n))f^\prime (v_n)v_n}{|x|^\beta }dx\\{} & {} \quad \le C_\mu \Bigg (\int _{\mathbb {R}^2}\frac{|h(f(v_n))f(v_n)|^{\frac{4}{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{2}}\\{} & {} \quad \le C_\mu \Bigg (\int _{\mathbb {R}^2}\frac{ |v_n|^{2}}{|x|^{\frac{4\beta }{4-\mu }}}dx \Bigg )^{\frac{4-\mu }{2}} +C \Bigg (\int _{\mathbb {R}^2}|x|^{-\frac{4\beta }{4-\mu }}|v_n|^{\frac{4q}{4-\mu }}\left( e^{\frac{8\alpha }{4-\mu }|v_n|^2}-1\right) dx\Bigg )^{\frac{4-\mu }{2}}\\{} & {} \quad \le C \Vert v_n\Vert ^{ 4-\mu } +C \Bigg (\int _{\mathbb {R}^2}|x|^{-\frac{4\beta }{4-\mu }}|v_n|^{\frac{4q\nu }{4-\mu }} dx\Bigg )^{\frac{4-\mu }{2\nu }}\\{} & {} \qquad \Bigg (\int _{\mathbb {R}^2}|x|^{-\frac{4\beta }{4-\mu }} \left( e^{\frac{8\alpha \nu ^\prime (1+\zeta )^2|\nabla v_n |_2^2}{4-\mu }(v_n/ (1+\zeta )|\nabla v_n |_2)^2}-1\right) dx\Bigg )^{\frac{4-\mu }{2\nu ^\prime }}\\{} & {} \quad \le C \Vert v_n\Vert ^{ 4-\mu }+C\Vert v_n\Vert ^{2q}\le C<+\infty , \end{aligned}$$

for $\zeta \approx 0^+$. In view of the Cauchy–Schwarz inequality in [32, 9.8 Theorem], or [9, Lemma 5.10], one obtains

$$\begin{aligned}{} & {} \Bigg |\int _{\mathbb {R}^2} \Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_n))}{|y|^\beta |x-y|^\mu }dy\Bigg )\frac{h(f(v_n))f^\prime (v_n)v_n}{|x|^\beta }dx\Bigg |\\{} & {} \quad \le \Bigg (\int _{\mathbb {R}^2} \Bigg (\int _{\mathbb {R}^2}\frac{H(f(v_n))}{|y|^\beta |x-y|^\mu }dy\Bigg )\frac{H(f(v_n))}{|x|^\beta }dx\Bigg )^{\frac{1}{2}} C^{\frac{1}{2}} \end{aligned}$$

which together with $\mathcal {J}_f^\prime (v_n)[v_n]\rightarrow 0$ yields that $\Vert v_n\Vert \rightarrow 0$ violating (2.27). The proof is finished. $\square$

Nevertheless, we have certified that $v_0\in E$ is a nontrivial critical point of $\mathcal {J}_f$, one could never draw the conclusion that $\mathcal {J}_f(v_0) = \inf _{v\in \mathcal {N}_f} \mathcal {J}_f(v)$. Indeed, without adopting the concentration-compactness principle in the Trudinger–Moser inequality sense developed by Lions [34], later generalized in [18], we would not even conclude that $\mathcal {J}_f(v_0)$ equals to the mountain-pass value c. For this aim, motivated by [9, Lemma 5.1], we introduce the following energy value associated with $\mathcal {J}_f$

$$\begin{aligned} c_1\triangleq \inf _{v\in X\backslash \{0\}}\max _{t>0}\mathcal {J}_f(tv). \end{aligned}$$

Moreover, we can prove that

Lemma 2.11

Let (V) and (1.8) with $(h_1)-(h_4)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, for all $v\in E\backslash \{0\}$, there is a unique $t_v>0$ such that $t_vv\in \mathcal {N}_f$ and then $c_2\triangleq \inf _{v\in \mathcal {N}_f} \mathcal {J}_f(v)=c_1\ge c$.

Proof

Define $\tau (t)\triangleq \mathcal {J}_f(tv)$ for all $t>0$, proceeding as the proof of Lemma 2.4, the existence of $t_v$ is trivial and we omit it here. Next, we concentrate ourself on how to show the uniqueness of $t_v$. It follows from [57, Theorem 4.1] that

$$\begin{aligned} \begin{aligned}&\tau ^\prime (t)=0 \Longleftrightarrow \mathcal {J}_f^\prime (t)(tv)[v]=0\\&\quad \Longleftrightarrow t \int _{\mathbb {R}^2}|\nabla v|^2dx+\int _{\mathbb {R}^2}V(x)f(tv)f^\prime (tv)vdx\\&\quad = \int _{\mathbb {R}^2} [ |x|^{-\mu }*|x|^{-\beta }H(f(tv))]|x|^{-\beta }h(f(tv))f^\prime (tv)v. \end{aligned} \end{aligned}$$

Then, thanks to (1.13) and (1.14) with $\delta \in [\frac{1}{2},1)$, we can claim that

$$\begin{aligned} h(f(s))f^\prime (s) ~\text {and}~H(f(s))/s~\text {are increasing on}~s\in (0,+\infty ). \end{aligned}$$

(2.29)

By $(f_9)$ in Lemma 2.1 and (2.29), it can infer that

$$\begin{aligned}{} & {} \int _{\mathbb {R}^2}V(x)f(tv)f^\prime (tv)(tv)^{-1}v^2dx \\{} & {} \quad -\int _{\mathbb {R}^2}\Bigg [\Bigg [ |x|^{-\mu }*\frac{|x|^{-\beta }H(f(tv))}{tv}v\Bigg ]\Bigg ]|x|^{-\beta }h(f(tv))f^\prime (tv)v \end{aligned}$$

is decreasing on $t\in (0,+\infty )$. Thus, we can deduce that $t_v$ is unique. To show $c_2=c_1$, we claim that

$$\begin{aligned} \mathcal {J}_f(v)-\mathcal {J}_f(t v)-\frac{1-t^2}{2t^2}\mathcal {J}^\prime _f(v)[v]\ge 0,~\forall t>0. \end{aligned}$$

(2.30)

To illustrate it, for all $t>0$ and $s,s_1,s_2>0$, we define

$$\begin{aligned} \left\{ \begin{array}{ll} \xi (t,s) \triangleq \frac{1}{2}|f(s)|^2-\frac{1}{2}|f(st)|^2-\frac{1-t^2}{2}f(s)f^\prime (s)s \\ \zeta (t,s_1,s_2) \triangleq \frac{1-t^2}{2}[H(f(s_2)) h(f(s_1))f^\prime (s_1)s_1+H(f(s_1)) h(f(s_2))f^\prime (s_2)s_2]\\ \quad +H(f(ts_2))H(f(ts_1)) -H(f(s_2))H(f(s_1)). \end{array} \right. \end{aligned}$$

It follows from some simple calculations that we can get (2.30) if $\xi (t,s)\ge 0$ and $\zeta (t,s_1,s_2)\ge 0$ for all $t>0$ and $s,s_1,s_2>0$. Firstly, with the help of $(f_9)$ in Lemma 2.1,

$$\begin{aligned} \frac{\partial }{\partial t} \xi (t,s)&=ts^2\big [ f(s)f^\prime (s)s^{-1}-f(st)f^\prime (st)(st)^{-1} \big ] \left\{ \begin{array}{ll} \ge 0, &{} \quad \text {if}~t\in [1,+\infty ), \\ <0, &{}\quad \text {if}~t\in (0,1], \end{array} \right. \end{aligned}$$

which revels that $\xi (t,s)\ge \min _{t>0}\xi (t,s)=\xi (1,s)=0$. Secondly, we exploit (2.29) to obtain

$$\begin{aligned} \frac{\partial }{\partial t}\zeta (t,s_1,s_2)&=H(f(ts_1))h(f(ts_2))f^\prime (ts_2)s_2 +H(f(ts_2))h(f(ts_1))f^\prime (ts_1)s_1\\&\quad -t[H(f(s_2)) h(f(s_1))f^\prime (s_1)s_1+H(f(s_1)) h(f(s_2))f^\prime (s_2)s_2]\\&=ts_1s_2\Bigg [\Bigg (\frac{H(f(ts_1))}{ts_1}h(f(ts_2))f^\prime (ts_2)-\frac{H(f(s_1))}{s_1}h(f(s_2))f^\prime (s_2)\Bigg )\\&\quad +\Bigg (\frac{H(f(ts_2))}{ts_2}h(f(ts_1))f^\prime (ts_1)-\frac{H(f(s_2))}{s_2}h(f(s_1))f^\prime (s_1)\Bigg ) \Bigg ]\\&\quad \left\{ \begin{array}{ll} \ge 0, &{}\quad \text {if}~t\in [1,+\infty ), \\ <0, &{}\quad \text {if}~t\in (0,1], \end{array} \right. \end{aligned}$$

and so $\zeta (t,s_1,s_2)\ge \min _{t>0}\zeta (t,s_1,s_2)=\zeta (1,s_1,s_2)=0$. With (2.30) in hand, we immediately have $c_2\ge c_1$. Alternatively, the existence of $t_v>0$ is sufficient to show that $c_2\le c_1$ and $c\le c_1$. Thereby, we can accomplish the proof of this lemma. $\square$

Now, we can present the proof of Theorem 1.3 as follows.

Proof of Theorem 1.3

The remainder is to show $c=c_2$. To the end, adopting $(f_6)$ and (1.13) with $\delta =\frac{1}{2}$,

$$\begin{aligned} c_1&\ge c=\liminf _{n\rightarrow \infty }\mathcal {J}_f(v_n)=\liminf _{n\rightarrow \infty } \left( \mathcal {J}_f(v_n)-\frac{1}{2}\mathcal {J}^\prime _f(v_n)[v_n]\right) \\&= \liminf _{n\rightarrow \infty }\Bigg [\frac{1}{2}\int _{\mathbb {R}^2}V(x)[|f(v_n)|^2-f(v_n)f^\prime (v_n)v_n]dx\\&\quad +\frac{1}{2}\int _{\mathbb {R}^2}\frac{H(f(v_n(y))) [h(f(v_n(x)))f^\prime (v_n(x))v_n(x)-H(f(v_n(x)))]}{|y|^\beta |x-y|^\mu |x|^\beta }dxdy\Bigg ]\\&\ge \mathcal {J}_f(v_0)-\frac{1}{2}\mathcal {J}^\prime _f(v_0)[v_0]=\mathcal {J}_f(v_0)\ge c_2=c_1 \end{aligned}$$

finishing the proof. $\square$

3 The proof of Theorem 1.7

In this section, to deal with Eq. (1.15) variationally, what the priority is how to verify that the variational functional I given by (1.16) is well-defined and continuous in $\mathcal {X}$. Since we consider Eq. (1.15) in $\mathcal {X}$ instead of $H^1(\mathbb {R}^2)$, Proposition 1.2 seems to be unapplicable to handle the nonlinearity with a critical exponential growth like (1.8). To begin with, we develop the following version type of Trudinger–Moser inequality.

Proposition 3.1

For all $\alpha >0$, $s\in [0,2)$ and $u\in \mathcal {X}$, there holds

$$\begin{aligned} |x|^{-s}(e^{\alpha u^4}-1)\in L^1(\mathbb {R}^2). \end{aligned}$$

(3.1)

Moreover, there exists a universal constant $C>0$, independent of u, such that

$$\begin{aligned} \sup _{u\in \mathcal {X}:d_\mathcal {X}(u,0)\le 1}\int _{\mathbb {R}^2}|x|^{-s}(e^{\alpha u^4}-1)dx\le C \end{aligned}$$

(3.2)

provided that $\frac{\alpha }{\pi }+\frac{s}{2}<1$.

Proof

Since $u\in \mathcal {X}$, then $u^2\in H^1(\mathbb {R}^2)$. If we replace $u^2$ with u in (1.10), we obtain (3.1) immediately. To show (3.2), one firstly finds that $\{u\in \mathcal {X}:d_\mathcal {X}(u,0)\le 1\}\subset \{u \in H^1(\mathbb {R}^2): \Vert u\Vert _{H^1(\mathbb {R}^2)}\le 1\}$. In fact, for every $u\in \{u\in \mathcal {X}:d_\mathcal {X}(u,0)\le 1\}$, the definition of $d_\mathcal {X}(u,0)$ reveals that $4\Vert u\Vert _{H^1(\mathbb {R}^2)}^2+|\nabla u^2|_2^2\le 4$ and so $\Vert u\Vert _{H^1(\mathbb {R}^2)}^2\le 1$. Moreover, due to the Ladyzhenskaya inequality (see e.g. [23, (II.3.9)]), we obtain

$$\begin{aligned} |u^2|_2^2=|u|_4^4\le 2^{-1}|u|_2^2|\nabla u|_2^2\le 2^{-1}(|u|_2^2+|\nabla u|_2^2)=2^{-1}\Vert u\Vert _{H^1(\mathbb {R}^2)}^2 \end{aligned}$$

which implies that

$$\begin{aligned} \Vert u^2\Vert _{H^1(\mathbb {R}^2)}^2=|u^2|_2^2+|\nabla u^2|_2^2\le 4\Vert u\Vert _{H^1(\mathbb {R}^2)}^2+|\nabla u^2|_2^2\le 4. \end{aligned}$$

Thereby, for all $u\in \{u\in \mathcal {X}:d_\mathcal {X}(u,0)<1\}$ and $\frac{\alpha }{\pi }+\frac{s}{2}<1$, then $\frac{\alpha \Vert u^2\Vert ^2_{H^1(\mathbb {R}^2)}}{4\pi }+\frac{s}{2}<1$. By (1.11),

$$\begin{aligned}{} & {} \int _{\mathbb {R}^2}|x|^{-s}(e^{\alpha u^4}-1)dx\\{} & {} \quad = \int _{\mathbb {R}^2}|x|^{-s}\Bigg (e^{\alpha \Vert u^2\Vert ^2_{H^1(\mathbb {R}^2)}\big ( (u^2)^2/\Vert u^2\Vert ^2_{H^1(\mathbb {R}^2)}\big ) }-1\Bigg )dx\le C<+\infty \end{aligned}$$

yielding the desired result. The proof of this proposition is complete. $\square$

Lemma 3.2

Suppose that h satisfies (1.8) and $(h_1)$, then the functional I in (1.16) is well-defined and continuous in $\mathcal {X}$. Moreover, if the Gateaux derivative of I evaluated in $u\in \mathcal {X}$ is zero in every direction $\psi \in C_0^\infty (\mathbb {R}^2)$, then u is a weak solution of Eq. (1.15).

Proof

Let $\varepsilon =1$ in (2.3) with suitable $\alpha$ and $\nu ^\prime$, by the HLS inequality and (2.3), for all $u\in \mathcal {X}$, we exploit (3.1) to get

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(u(x))H(u(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx \le C_\mu \Bigg (\int _{\mathbb {R}^2}\frac{|H(u)|^{\frac{4}{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{2}}\\&\quad \le C_\mu \Bigg (\int _{\mathbb {R}^2}\frac{ |u|^{2}}{|x|^{\frac{4\beta }{4-\mu }}}dx \Bigg )^{\frac{4-\mu }{2}} +C \Bigg (\int _{\mathbb {R}^2} \frac{|u|^{\frac{4q}{4-\mu }}\left( e^{\frac{4\alpha }{4-\mu }|u|^4}-1\right) dx}{|x|^{\frac{4\beta }{4-\mu }}}\Bigg )^{\frac{4-\mu }{2}}\\&\quad \le C \Vert u\Vert ^{ 4-\mu } +C \Bigg (\int _{\mathbb {R}^2}\frac{|u|^{\frac{4q\nu }{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}} dx\Bigg )^{\frac{4-\mu }{2\nu }} \Bigg (\int _{\mathbb {R}^2}|x|^{-\frac{4\beta }{4-\mu }}\left( e^{\frac{4\alpha }{4-\mu }|u|^4}-1\right) dx\Bigg )^{\frac{4-\mu }{2\nu ^\prime }} <+\infty \end{aligned} \end{aligned}$$

implying that $I:\mathbb {R}\rightarrow \mathbb {R}$ is well-defined, where we have used Lemma 2.3 since $\frac{4\beta }{4-\mu }\in (0,2)$.

Next, we derive that $I:\mathbb {R}\rightarrow \mathbb {R}$ is continuous on $\mathcal {X}$. Let us suppose that $d_\mathcal {X}(u_n,u)\rightarrow 0$ indicating that $u_n\rightarrow u$ in $H^1(\mathbb {R}^2)$ and $|\nabla u_n^2|_2^2\rightarrow |\nabla u^2|_2^2$. Furthermore, one has $u_n^2\rightarrow u^2$ in $H^1(\mathbb {R}^2)$. Proceeding as the proof of [44, Proposition 1], there are a subsequence $\{u_n\}$ still denoted by itself and a function $v\in \mathcal {X}$ such that $|u_n(x)|\le v(x)$ a.e. in $\mathbb {R}^2$. So, we can deduce that there is a constant $C=C(v)>0$ independent of $n\in \mathbb {N}$ such that

$$\begin{aligned} \int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{h(u_n(x))u_n(x)H(u_n(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx\le C. \end{aligned}$$

Repeating the very similar arguments explored in Lemma 2.9, we have that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(u_n(x)) H(u_n(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx= \int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(u(x)) H(u(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx \end{aligned}$$

implying that $I(u_n)\rightarrow I(u)$. So, $I\in C^0(\mathcal {X},\mathbb {R})$. The remaining part of this lemma is trivial and we omit it here, the interested reader can refer to [57]. We finish the proof of this lemma. $\square$

Lemma 3.3

Suppose that h satisfies (1.8) and $(h_1)$, then every nontrivial nontrivial solution u of Eq. (1.15) belongs to $\mathcal {M}=\{u\in \mathcal {X}\backslash \{0\}:P(u)=0\}$, where

$$\begin{aligned} P(u)\triangleq 2\int _{\mathbb {R}^2} |u|^2dx -(4-2\beta -\mu )\int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(u(x))H(u(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx,~\forall u\in \mathcal {X}\backslash \{0\}. \end{aligned}$$

Proof

Although there is a quasilinear term in Eq. (1.15), we could follow the proof of [9, Theorem 1.6] to obtain that $u\in W^{2,t}(\mathbb {R}^2)$ for every $t\in ((1-\epsilon _0)s,s)$ with $\epsilon _0>0$ being sufficiently small and $s \in (\frac{2}{\beta +\mu },\frac{2}{\beta })$. With this fact in hand, it is standard to show that $P(u)\equiv 0$, see e.g. [9, Theorem 1.7] in detail. The proof is complete. $\square$

Now, we turn to focus on verifying the necessary properties for the manifold $\mathcal {M}$.

Lemma 3.4

Suppose that h satisfies (1.8), $(h_1)$ and $(h_5)$. If $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, then for every $u\in \mathcal {X}\backslash \{0\}$, there exists a unique $t_u > 0$ such that $u_{t_u}=u(t_u^{-1}\cdot )\in \mathcal {M}$. In particular, $I(u_{t_u})=\max _{t>0}I(u_t)$ and then

$$\begin{aligned} c_\mathcal {M}\triangleq \inf _{u\in \mathcal {M}}I(u)=\inf _{u\in \mathcal {X}\backslash \{0\}}\max _{t>0}I(u_t). \end{aligned}$$

(3.3)

Proof

In consideration of $|\nabla u^2_t|_2=|\nabla u^2|_2$ for every $t>0$ and Lemma 3.2, it is sufficient to arguing as the proof of [9, Lemma 4.9] to finish the proof of this lemma. $\square$

Lemma 3.5

Suppose that h satisfies (1.8), $(h_1)$ and $(h_5)$. If $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, then there is a $\xi _0>0$ such that for all $\xi >\xi _0$, there holds

$$\begin{aligned}c_\mathcal {M}<c_*\triangleq \min \Bigg \{ \frac{\pi (2-2\beta -\mu )}{8\alpha _0}, \frac{2-2\beta -\mu }{ 2(4-2\beta -\mu )} \Bigg \}. \end{aligned}$$

Proof

Let $\varphi _0\in \mathcal {X}$ be a cut-off function satisfying $\varphi _0\in C_{0}^\infty (\mathbb {R}^2)$ defined by $0\le \varphi _0(x)\le 1$ for every $x\in \mathbb {R}^2$, $\varphi _0(x)\equiv 1$ if $|x|\le 1/2$, $\varphi _0(x)\equiv 0$ if $|x|\ge 1$ and $|\nabla \varphi _0|\le 1$ for all $x\in \mathbb {R}^2$. So, for all $\theta ,t>0$,

$$\begin{aligned} I((\theta \varphi _0)_t)&=\frac{1}{2}\int _{\mathbb {R}^2}[(1+ |\theta \varphi _0|^2) |\theta \nabla \varphi _0|^2 +t^2|\theta \varphi _0|^2]dx\nonumber \\&\quad -\frac{t^{4-2\beta -\mu }}{2} \int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(\theta \varphi _0(x))H(\theta \varphi _0(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx\nonumber \\&\le \frac{1}{2}\int _{\mathbb {R}^2}[(1+\theta ^2|\varphi _0|^2)\theta ^2|\nabla \varphi _0|^2 +t^2\theta ^2 |\varphi _0|^2dx\nonumber \\&\quad - \frac{t^{4-2\beta -\mu }}{2}C_\mu \xi ^2\theta ^{2p}\Bigg (\int _{B_1(0)}|\varphi _0|^pdx\Bigg )^2\nonumber \\&\le \frac{1}{2}\int _{\mathbb {R}^2}[(1+\theta ^2|\varphi _0|^2)\theta ^2|\nabla \varphi _0|^2dx+ \frac{\pi }{2} t^2 \theta ^2 \nonumber \\&\quad -\frac{t^{4-2\beta -\mu }}{2}C_\mu \xi ^2\theta ^{2p}\Bigg (\int _{B_1(0)}|\varphi _0|^pdx\Bigg )^2, \end{aligned}$$

(3.4)

where $C_\mu >0$ is a constant dependent of $\mu$. We choose $\theta =\min \{\sqrt{\frac{c_*}{2\pi +1}},1\}>0$, and $|\nabla \varphi _0|_2^2\le \pi$ as well as $|\varphi _0|\le 1$, by (3.4), we get

$$\begin{aligned} I((\theta \varphi _0)_t)&\le \frac{1}{2}\int _{\mathbb {R}^2}[(1+\theta ^2|\varphi _0|^2)\theta ^2|\nabla \varphi _0|^2dx + \frac{\pi }{2} t^2 \theta ^2 -\frac{t^{4-2\beta -\mu }}{2}C_\mu \xi ^2\theta ^{2p}|\varphi _0|^{2p}_{L^p(B_1(0))}\nonumber \\&\le \frac{\pi c_*}{2\pi +1}+ \frac{\pi }{2} t^2 \theta ^2 -\frac{t^{4-2\beta -\mu }}{2}C_\mu \xi ^2\theta ^{2p}|\varphi _0|^{2p}_{L^p(B_1(0))}. \end{aligned}$$

(3.5)

It infers from some elementary calculations that

$$\begin{aligned}{} & {} \max _{t>0}\Bigg ( \frac{\pi }{2} t^2 \theta ^2 -\frac{t^{4-2\beta -\mu }}{2}C_\mu \xi ^2\theta ^{2p}|\varphi _0|^{2p}_{L^p(B_1(0))}\Bigg )\nonumber \\{} & {} \qquad =\frac{(2-2\beta -\mu )\pi \theta ^2}{ 2(4-2\beta -\mu )}\Bigg ( \frac{2\pi }{(4-2\beta -\mu )C_\mu \xi ^2\theta ^{2(p-1)}|\varphi _0|^{2p}_{L^p(B_1(0))}} \Bigg )^{\frac{2}{2-2\beta -\mu }}<\frac{c_*}{2} \nonumber \\ \end{aligned}$$

(3.6)

provided that $\xi >\xi _0$ with $\xi _0$ defined by

$$\begin{aligned} \xi _0\triangleq \Bigg (\frac{ (2-2\beta -\mu )\pi \theta ^2}{c_* (4-2\beta -\mu )}\Bigg ) ^{\frac{2-2\beta -\mu }{4}}\sqrt{\frac{2\pi }{(4-2\beta -\mu )C_\mu \theta ^{2(p-1)}|\varphi _0|^{2p}_{L^p(B_1(0))}}}>0. \end{aligned}$$

Combining (3.3) and (3.5)–(3.6), we conclude the proof of this lemma. $\square$

Lemma 3.6

Suppose that h satisfies (1.8), $(h_1)$ and $(h_5)$. If $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$ then there is a constant $\varrho >0$ such that $|u_n|_2\ge \varrho$ for all $n\in \mathbb {N}$, where $\{u_n\}\subset \mathcal {M}$ is a minimizing sequence of $c_\mathcal {M}$. In particular, we have that $c_{\mathcal {M}}> 0$.

Clearly, $(h_1)$ indicates the condition $\lim _{s\rightarrow 0}h(s)/s^{2-2\beta -\mu }=0$. We then introduce the following type Trudinger–Moser inequality due to Adachi–Tanaka [2]

$$\begin{aligned} \int _{\mathbb {R}^2}\big (e^{\alpha v^{2}/ |\nabla v |_{2}^2}-1\big )dx\le C \frac{ |v |_{2}^2}{|\nabla v|_{2}^2}~\text{ if }~\alpha <4\pi ,~\forall v\in H^{1}(\mathbb {R}^2)\backslash \{0\}, \end{aligned}$$

(3.7)

in the space $\mathcal {X}$. It follows from this condition and (1.8) that for all fixed $\alpha >\alpha _0$ and $\overline{q}\ge 4-2\beta -\mu$,

$$\begin{aligned} |H(s)|\le |s|^{\frac{4-2\beta -\mu }{2}}+C(\alpha ,\overline{q})|s|^{\overline{q}}(e^{\alpha s^4}-1),~\forall s\in \mathbb {R}. \end{aligned}$$

(3.8)

Moreover, in consideration of the critical exponential growth (1.8), we must modify some computations used in [9, Lemma 4.10]. For this purpose, we introduce the following sharp Gagliardo-Nirenberg inequality [6]

$$\begin{aligned} \int _{\mathbb {R}^2}|v|^{\frac{r}{2}}dx\le C\Bigg (\int _{\mathbb {R}^2}|\nabla v|^{2}dx\Bigg )^{\frac{r-2}{4}}\int _{\mathbb {R}^2}|v| dx,~\forall r>2 ~\text {and}~v\in D^{2,1}(\mathbb {R}^2), \nonumber \\ \end{aligned}$$

(3.9)

where $D^{2,1}(\mathbb {R}^2)=\{v\in L^1(\mathbb {R}^2):|\nabla v|\in L^2(\mathbb {R}^2)\}$.

Now, we can show the proof of Lemma 3.6.

Proof

Suppose, by contradiction, that there is a subsequence, still denoted by itself, $\{u_n\}\subset \mathcal {M}$ such that $|u_n|_2\rightarrow 0$ in $H^1(\mathbb {R}^2)$. According to the definition of $\mathcal {M}$, then $[|x|^{-\mu }*(|x|^{-\beta }H(u_n))]|x|^{-\beta }H(u_n)\rightarrow 0$ in $L^1(\mathbb {R}^2)$ which implies that $\limsup _{n\rightarrow \infty }|\nabla u_n^2|_2^2\le 8c_{\mathcal {M}} <8c_*$. Hence, by Lemma 3.5, we shall choose $\alpha >\alpha _0$ sufficiently close to $\alpha _0$ and $\nu ^\prime >1$ sufficiently close to 1 in such a way that $1/v+1/v^\prime =1$ and

$$\begin{aligned} \frac{4\alpha \nu ^\prime |\nabla u_n^2|_2^2}{4-2\beta -\mu }<4\pi (1-\epsilon ) ~\text {for some suitable}~\epsilon \in (0,1). \end{aligned}$$

(3.10)

Let $v=u_n^2$ in (3.7) and (3.9), respectively. As a consequence of (3.8) and (3.10), we derive

$$\begin{aligned}{} & {} \frac{2}{4-2\beta -\mu }\int _{\mathbb {R}^2} |u_n|^2dx = \int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(u_n(x))H(u_n(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx\\{} & {} \quad \le C_{\beta ,\mu }\Bigg (\int _{\mathbb {R}^2}|H(u_n)|^{\frac{4}{4-2\beta -\mu }}dx\Bigg )^{\frac{4-2\beta -\mu }{2}} \\{} & {} \quad \le C_{\beta ,\mu }|u_n|_2^{4-2\beta -\mu }+C \Bigg (\int _{\mathbb {R}^2}|u_n|^{\frac{4\overline{q}}{4-2\beta -\mu }}\left( e^{ \frac{4\alpha }{4-2\beta -\mu }|u_n|^4}-1\right) dx\Bigg )^{\frac{4-2\beta -\mu }{2}}\\{} & {} \quad \le C_{\beta ,\mu } |u_n|_2^{4-2\beta -\mu }+C \Bigg (\int _{\mathbb {R}^2}|u_n|^{\frac{4\overline{q}\nu }{4-2\beta -\mu }}dx\Bigg )^{\frac{4-2\beta -\mu }{2\nu }}\\{} & {} \qquad \Bigg (\int _{\mathbb {R}^2}\left( e^{ \frac{4\alpha \nu ^\prime |\nabla u_n^2|_{2}^2 }{4-2\beta -\mu } (|u_n^2|^2/|\nabla u_n^2|^2_{2}) }-1\right) dx\Bigg )^{\frac{4-2\beta -\mu }{2\nu ^\prime }}\\{} & {} \quad \le C_{\beta ,\mu }|u_n|_2^{4-2\beta -\mu }+C |u_n|_2^{\frac{4-2\beta -\mu }{\nu }}|\nabla u_n^2|_2^{ \overline{q}-\frac{4-2\beta -\mu }{2\nu }} \Bigg (\frac{|u_n^2|_2}{|\nabla u_n^2|_2}\Bigg )^{\frac{4-2\beta -\mu }{\nu ^\prime }}\\{} & {} \quad \le C_{\beta ,\mu }|u_n|_2^{4-2\beta -\mu }+ C|u_n|_2^{4-2\beta -\mu }|\nabla u_n|_2^{\frac{4-2\beta -\mu }{\nu ^\prime }}|\nabla u_n^2|_2^{\frac{4-2\beta -\mu }{2\nu }}\\{} & {} \quad \le (C_{\beta ,\mu }+C)|u_n|_2^{4-2\beta -\mu } \end{aligned}$$

yielding that $|u_n|_2\ge C>0$, where C is a constant independent of $n\in \mathbb {N}$, where we have adopted the Ladyzhenskaya inequality in the last second inequality. This is impossible because $|u_n|_2\rightarrow 0$. The remaining part is trivial and we omit it. The proof of this lemma is finished. $\square$

Lemma 3.7

Suppose that h satisfies (1.8), $(h_1)$ and $(h_5)$. If $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$ then the minimization problem $c_{\mathcal {M}}=\inf _{u\in \mathcal {M}}I(u)$ would be achieved.

Proof

Let $\{u_n\}\subset \mathcal {M}$ be a minimizing sequence of $c_\mathcal {M}$, that is, $I(u_n)\rightarrow c_\mathcal {M}$ and $P(u_n)=0$. Hence,

$$\begin{aligned} c_\mathcal {M}+o_n(1)&=I(u_n)-\frac{1}{4-2\beta -\mu }P (u_n) \\&=\frac{1}{2}\int _{\mathbb {R}^2}[(1+ u_n^2)]|\nabla u_n|^2dx +\frac{ 2-2\beta -\mu }{2(4-2\beta -\mu )}\int _{\mathbb {R}^2}|u_n|^2dx\\&\ge \frac{ 2-2\beta -\mu }{2(4-2\beta -\mu )}\int _{\mathbb {R}^2}\big ([(1+ u_n^2)]|\nabla u_n|^2+|u_n|^2\big )dx\\&=\frac{ 2-2\beta -\mu }{2(4-2\beta -\mu )}d^2_\mathcal {X}(u_n,0) \end{aligned}$$

revealing that $\{u_n\}$ is uniformly bounded in $\mathcal {X}$. In fact, recalling Lemma 3.5, we can further obtain

$$\begin{aligned} \limsup _{n\rightarrow \infty }d^2_\mathcal {X}(u_n,0)\le \frac{ 2(4-2\beta -\mu )}{2-2\beta -\mu }c_\mathcal {M} <\min \Bigg \{ \frac{\pi (4-2\beta -\mu )}{4\alpha _0},1\Bigg \}. \end{aligned}$$

(3.11)

Going to a subsequence if necessary, there is a function $u\in \mathcal {X}$ such that $u_n\rightharpoonup u$ in $H^1(\mathbb {R}^2)$, $u_n^2\rightharpoonup u^2$ in $H^1(\mathbb {R}^2)$, $u_n \rightarrow u$ in $L^q(\mathbb {R}^2,|x|^{-s}dx)$ for all $s\in (0,2)$ with $q\in [2,+\infty )$, and $u_n\rightarrow u$ a.e. in $\mathbb {R}^2$. To conclude that $d_\mathcal {X}(u_n,u)\rightarrow 0$, we claim that

$$\begin{aligned}{} & {} \lim _{n\rightarrow \infty }\int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(u_n(x))H(u_n(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx\nonumber \\{} & {} \quad = \int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(u (x))H(u (y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx. \end{aligned}$$

(3.12)

Indeed, by means of $d_\mathcal {X}(u_n,0)<1$ for all sufficiently large $n\in \mathbb {N}$ in (3.11), we have that

$$\begin{aligned} d_\mathcal {X}\Bigg (\frac{u_n}{\sqrt{d_\mathcal {X}(u_n,0)}},0\Bigg )= & {} \frac{1}{2} \sqrt{\frac{4\Vert u_n\Vert ^2_{H^1(\mathbb {R}^2)}}{d_\mathcal {X}(u_n,0)}+\frac{|\nabla u_n^2|_2^2}{d^2_\mathcal {X}(u_n,0)}} \\\le & {} \frac{ \sqrt{4\Vert u_n\Vert ^2_{H^1(\mathbb {R}^2)}+|\nabla u_n^2|_2^2} }{2d_\mathcal {X}(u_n,0)}=1. \end{aligned}$$

For the fixed $\alpha >\alpha _0$, $q\ge \frac{4-\mu }{2\nu }$ with $1/\nu +1/\nu ^\prime =1$ in (2.3), letting $\varepsilon =1$ with suitable $\alpha$ and $\nu ^\prime$, there holds $\frac{4\alpha \nu ^\prime d^2_\mathcal {X}(u_n,0)}{\pi }+\frac{s}{2}<1$ with $s=\frac{4\beta }{4-\mu }\in (0,2)$, we can apply (3.2) with denoting $v_n=\frac{u_n}{\sqrt{d_\mathcal {X}(u_n,0)}}$ (which gives that $d_\mathcal {X}(v_n,0)\le 1$) to obtain for each measurable set $\Omega \subset \mathbb {R}^2$ the inequality below

$$\begin{aligned}{} & {} \int _{\Omega }\int _{\mathbb {R}^2}\frac{H(u_n(x))H(u_n(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx\\{} & {} \quad \le C_\mu \Bigg (\int _{\Omega }\frac{|H(u_n)|^{\frac{4}{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{4}} \Bigg (\int _{\mathbb {R}^2}\frac{|H(u_n)|^{\frac{4}{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{4}}\\{} & {} \quad \le C \Bigg (\int _{\Omega }\frac{|H(u_n)|^{\frac{4}{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{4}}\\{} & {} \quad \le C\Bigg (\int _{\Omega }\frac{ |u_n|^{2}}{|x|^{\frac{4\beta }{4-\mu }}}dx \Bigg )^{\frac{4-\mu }{4}} +C \Bigg (\int _{\Omega } \frac{|u_n|^{\frac{4q}{4-\mu }}\left( e^{\frac{4\alpha }{4-\mu }|u_n|^4}-1\right) dx}{|x|^{\frac{4\beta }{4-\mu }}}\Bigg )^{\frac{4-\mu }{4}}\\{} & {} \quad \le C\Bigg (\int _{\Omega }\frac{ |u_n|^{2}}{|x|^{\frac{4\beta }{4-\mu }}}dx \Bigg )^{\frac{4-\mu }{4}} +C \Bigg (\int _{\Omega }\frac{|u_n|^{\frac{4q\nu }{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}} dx\Bigg )^{\frac{4-\mu }{4\nu }}\\{} & {} \qquad \Bigg (\int _{\mathbb {R}^2}|x|^{-\frac{4\beta }{4-\mu }} \left( e^{\frac{4\alpha \nu ^\prime d^2_\mathcal {X}(u_n,0)}{4-\mu } v_n^4}-1\right) dx\Bigg )^{\frac{4-\mu }{4\nu ^\prime }}\\{} & {} \quad \le C\Bigg (\int _{\Omega }\frac{ |u_n|^{2}}{|x|^{\frac{4\beta }{4-\mu }}}dx \Bigg )^{\frac{4-\mu }{4}} +C \Bigg (\int _{\Omega }\frac{|u_n|^{\frac{4q\nu }{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}} dx\Bigg )^{\frac{4-\mu }{4\nu }}. \end{aligned}$$

Therefore, setting the numbers $l_1=\frac{4-\mu }{4}$, $l_2=\frac{4-\mu }{4\nu }$ and the sequences

$$\begin{aligned} f_n\triangleq \int _{\mathbb {R}^2}\frac{H(u_n(x))H(u_n(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dy, \quad g_n \triangleq \frac{|u_n|^{2}}{|x|^{\frac{4\beta }{4-\mu }}} \quad \text{ and } \quad h_n\triangleq \frac{|u_n|^{\frac{4q\nu }{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}}, \end{aligned}$$

we have that

$$\begin{aligned}{} & {} f_n \rightarrow \int _{\mathbb {R}^2}\frac{H(u(x))H(u(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dy\triangleq f~\text {a.e.}~\text {in}~\mathbb {R}^2,\\{} & {} g_n \rightarrow \frac{|u|^{2}}{|x|^{\frac{4\beta }{4-\mu }}} \triangleq g \quad \text{ in } \quad L^{1}(\mathbb {R}^2), \quad h_n \rightarrow \frac{|u|^{\frac{4q\nu }{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}} \triangleq h \quad \text{ in } \quad L^{1}(\mathbb {R}^2), \end{aligned}$$

and

$$\begin{aligned} \int _{\Omega }f_ndx\le \left( \int _{\Omega }g_ndx\right) ^{l_1}+\left( \int _{\Omega }h_ndx\right) ^{l_2}, \end{aligned}$$

for all measurable set $\Omega \subset \mathbb {R}^2$. This jointly with the Vitali’s Dominated Convergence theorem to get (3.12). With the help of (3.12) and the Fatou’s lemma,

$$\begin{aligned} P(u)\le \liminf _{n\rightarrow \infty } P(u_n)=0. \end{aligned}$$

(3.13)

Combining the first part of Lemma 3.6 and (3.12), we immediately conclude that $u\ne 0$. Therefore, by means of (3.13), there exists a constant $t_0\in (0,1]$ such that $P(u_{t_0})=0$. Using Fatou’s lemma again, we obtain

$$\begin{aligned} c_\mathcal {M}&\le I(u_{t_0})=I(u_{t_0})-\frac{1}{4-2\beta -\mu }P (u_{t_0})\\&= \frac{1}{2}\int _{\mathbb {R}^2}[(1+ u^2)]|\nabla u|^2dx +\frac{ 2-2\beta -\mu }{2(4-2\beta -\mu )}t_0^2\int _{\mathbb {R}^2}|u|^2dx\\&\le \frac{1}{2}\int _{\mathbb {R}^2}[(1+ u^2)]|\nabla u|^2dx +\frac{ 2-2\beta -\mu }{2(4-2\beta -\mu )} \int _{\mathbb {R}^2}|u|^2dx\\&\le \liminf _{n\rightarrow \infty } \Bigg (\frac{1}{2}\int _{\mathbb {R}^2}[(1+ u_n^2)]|\nabla u_n|^2dx +\frac{ 2-2\beta -\mu }{2(4-2\beta -\mu )}\int _{\mathbb {R}^2}|u_n|^2dx\Bigg )\\&=\liminf _{n\rightarrow \infty } \left[ I(u_n)-\frac{1}{4-2\beta -\mu }P (u_n) \right] =c_\mathcal {M} \end{aligned}$$

indicating the desired result. The proof of this lemma is complete. $\square$

We then try to certify that every attained function of the minimization problem $c_{\mathcal {M}}=\inf _{u\in \mathcal {M}}I(u)$. Before proceeding this aim, we have two claims listed as follows.

Claim A. For all $u\in \mathcal {X}$, there holds $d_{\mathcal {X}}(u_t,u)\rightarrow 0$ as $t\rightarrow 1$.

Verification: Since $C_0^\infty (\mathbb {R}^2)$ is dense in $H^1(\mathbb {R}^2)$, for each $\varepsilon >0$, there exist $U \in C^\infty _0(\mathbb {R}^2,\mathbb {R}^2)$ and $v \in C^\infty _0(\mathbb {R}^2)$ such that

$$\begin{aligned} \int _{\mathbb {R}^2}|\nabla u-U|^2dx<\frac{\varepsilon }{20} ~\text {and}~ \int _{\mathbb {R}^2}|u-v|^2dx<\frac{\varepsilon }{20}. \end{aligned}$$

By some simple computations

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^2}|\nabla u(t^{-1}x)-\nabla u(x) |^2dx\\&\quad = \int _{\mathbb {R}^2}| \nabla u(t^{-1}x)-t^{-1}U(t^{-1}x)+t^{-1}U(t^{-1}x)\\&\qquad -U(t^{-1}x)+U(t^{-1}x)-U (x)+U (x)- \nabla u(x) |^2dx\\&\quad \le 8 \int _{\mathbb {R}^2}| \nabla u -U |^2dx+4(t^{-1}-1)^2t^2 \int _{\mathbb {R}^2}|U|^2dx +4 \int _{\mathbb {R}^2}|U(t^{-1}x)-U (x)|^2dx \end{aligned} \end{aligned}$$

and

$$\begin{aligned}{} & {} \int _{\mathbb {R}^2}|u(t^{-1}x)- u(x)|^2dx\\{} & {} \quad =\int _{\mathbb {R}^2}|u(t^{-1}x)- v( t^{-1}x)+v( t^{-1}x) -v(x)+v(x)- u(x)|^2dx\\{} & {} \quad \le 3\int _{\mathbb {R}^2}| u(t^{-1} x)- v(t^{-1} x)|^2dx +3\int _{\mathbb {R}^2}| v(t^{-1} x)-v(x)|^2dx\\{} & {} \qquad +3\int _{\mathbb {R}^2}| v(x)- u( x) |^2dx \\{} & {} \quad \le 3(t^2+1)\int _{\mathbb {R}^2}| u(x)- v(x)|^2dx +3\int _{\mathbb {R}^2}| v(t^{-1} x)-v(x)|^2dx. \end{aligned}$$

Recalling that $|\nabla u_t^2|_2^2=|\nabla u^2|_2^2$ for all $t>0$, by letting $t\rightarrow 1$, we deduce that

$$\begin{aligned} \limsup _{t\rightarrow 1}d^2_{\mathcal {X}}(u_t,u)\le 8\int _{\mathbb {R}^2}| \nabla u(x)-U(x)|^2dx+ 8\int _{\mathbb {R}^2}| u( x)- v( x)|^2dx<\varepsilon \end{aligned}$$

showing the claim.

Claim B. By Lemma 3.4, the map $\mathcal {X}\backslash \{0\}\rightarrow (0,+\infty )$ defined by $u\mapsto t_u$ is continuous.

Verification: To finish it, we firstly have the following assertion:

$$\begin{aligned} \forall \{u_n\}\subset \mathcal {X} \backslash \{0\}~ \text {with} ~ d_\mathcal {X}(u_n,u_0) \rightarrow 0~ \text {and}~ u_0\in \mathcal {M}\Longrightarrow t_{n}\rightarrow 1, \end{aligned}$$

(3.14)

where $t_n\triangleq t_{u_n}>0$ satisfies $(u_n)_{t_n}=u_n(t_n^{-1}\cdot )\in \mathcal {M}$ (Lemma 3.4 derives the existence of $t_n$. For each fixed $t>0$, thanks to [44, Proposition 1], we can follow the proof of Lemma 3.2 to derive

$$\begin{aligned}{} & {} \lim _{n\rightarrow \infty }\int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(u_n(x)) H(u_n(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx\nonumber \\{} & {} \quad = \int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(u(x)) H(u(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx. \end{aligned}$$

(3.15)

Given $\varepsilon \in (0,1)$, define $s_1=1-\varepsilon$ and $s_2=1+\varepsilon$. Since $u_0\in \mathcal {M}$, then Lemma 3.4 reveals that $t=1$ is the unique maximum point of $I((u_0)_t)$, and therefore $\frac{d}{dt}|_{t=s_1}I((u_0)_{t})>0$ and $\frac{d}{dt}|_{t=s_2}I((u_0)_{t})<0$. So, by using (3.15), for some sufficiently large $n\in \mathbb {N}$, one has

$$\begin{aligned} \frac{d}{dt}\Bigg |_{t=s_1}I((u_n)_{t})>0~\text {and}~\frac{d}{dt}\Bigg |_{t=s_2}I((u_n)_{t})<0 \end{aligned}$$

which indicate that $1-\varepsilon<t_n<1+\varepsilon$. So, (3.14) holds true.

Let $\{u_n\}\subset \mathcal {X}\backslash \{0\}$ be a sequence satisfying $d_\mathcal {X}(u_n,u_0)$ as $n\rightarrow \infty$, there is a unique $t_u>0$ such that $u_{t_u}\in \mathcal {M}$. Denoting $v_n\triangleq (u_n)_{t_u}$, then $d_\mathcal {X}(v_n,u_{t_u})$ as $n\rightarrow \infty$. In view of (3.14), we have $\overline{t}_n\rightarrow 1$ as $n\rightarrow \infty$, where $(v_n)_{\overline{t}_n}\in \mathcal {M}$. On the other hand, it is easy to prove that, passing to a subsequence if necessary, $t_n\rightarrow t_0$ as $n\rightarrow \infty$, where $(u_n)_{t_n}\in \mathcal {M}$. By the uniqueness in Lemma 3.4, one deduces $\overline{t}_nt_u=t_n$ which indicates that $t_0=t_u$ by tending $n\rightarrow \infty$ on both sides. So, the claim is true.

Lemma 3.8

Suppose that h satisfies (1.8), $(h_1)$ and $(h_5)$. If $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, then any minimizer of $c_{\mathcal {M}}=\inf _{u\in \mathcal {M}}I(u)$ is a solution of Eq. (1.15).

Proof

Arguing it indirectly, we could suppose that u is not a weak solution of Eq. (1.15) and there exists a function $\varphi \in C_{0}^\infty (\mathbb {R}^2)$ such that $I^\prime (u)[\varphi ]<0$. Fixing $\epsilon >0$ such that

$$\begin{aligned} \begin{aligned}&\forall \vartheta \in (0,\epsilon ),~ \forall w\in \mathcal {X}~ \text {with}~ d_\mathcal {X}(w,0)<\epsilon ,~ \forall \overline{\varphi } ~ \text {with}~ d_\mathcal {X}(\overline{\varphi },\varphi )<\epsilon , \\&\quad \Longrightarrow I(u+w+\vartheta \overline{\varphi })\le I(u+w)-\epsilon \vartheta . \end{aligned} \end{aligned}$$

(3.16)

Due to Claim A, there exists a constant $\delta =\min \{\delta (u,\epsilon ), \delta (\varphi ,\epsilon )\}>0$ such that

$$\begin{aligned} |t-1|<\delta ~ \Longrightarrow ~ d_{\mathcal {X}}(u_t,u)<\epsilon ~ \text {and}~ d_{\mathcal {X}}(\varphi _t,\varphi ) <\epsilon . \end{aligned}$$

(3.17)

Set $Q^\vartheta \triangleq u+\vartheta \varphi \in \mathcal {X}$, thereby there exists a unique $t_{Q^\vartheta }>0$ such that $Q^\vartheta _{t_{Q^\vartheta }}\in \mathcal {M}$ by Lemma 3.4. As $u\in \mathcal {M}$, one has that $t_u=1$ by Lemma 3.4. Obviously, $d_{\mathcal {X}}(Q^\vartheta ,u) \rightarrow 0$ as $\vartheta \rightarrow 0$, hence $t_{Q^\vartheta }\rightarrow 1$ as $\vartheta \rightarrow 0$ by Claim B. For $\vartheta \in (0,\epsilon )$ sufficiently small, we can derive that $|t_{Q^\vartheta }-1|<\delta$ which together with (3.17) gives that

$$\begin{aligned} d_{\mathcal {X}}(u_{t_{Q^\vartheta }},u)<\epsilon ~ \text {and}~ d_{\mathcal {X}}(\varphi _{t_{Q^\vartheta }},\varphi )<\epsilon . \end{aligned}$$

(3.18)

Consequently, by (3.18) we may define $w\triangleq u_{t_{Q^\vartheta }}-u$ and $\overline{\varphi }\triangleq \varphi _{t_{Q^\vartheta }}$ in (3.16),

$$\begin{aligned} I(Q^\vartheta _{t_{Q^\vartheta }})&=I\big ( u_{t_{Q^\vartheta }} +\vartheta \varphi _{t_{Q^\vartheta }} \big ) =I(u+w+\vartheta \overline{\varphi })\le I(u+w)-\epsilon \vartheta \\&<I(u+w)=I(u_{t_{Q^\vartheta }})\le \max _{t>0}I(u_t)=I(u)=c_{\mathcal {M}}, \end{aligned}$$

which together with $Q^\vartheta _{t_{Q^\vartheta }}\in \mathcal {M}$ yields a contradiction. The proof is complete. $\square$

We are in a position to show the proof of Theorem 1.7.

Proof of Theorem 1.7

The set $\mathcal {M}$ is well-defined by Lemma 3.4 and then we could consider the minimization problem $c_\mathcal {M}=\inf _{u\in \mathcal {M}}I(u)$. Combining Lemmas 3.7 and 3.8, one concludes that the minimizer, say it $u\in \mathcal {X}$, is indeed a nontrivial solution of Pohoz̆aev type ground state of Eq. (1.15). According to (3.3), we can finish the proof. $\square$

4 The proofs of Theorems 1.8 and 1.11

In this section, we are mainly concerned with the case $\kappa >0$ for Eq. (1.1), where the existence of nontrivial solutions and the asymptotical behavior are investigated. Similar to Sect. 2, we have to determine a suitable work space and then consider the problems variationally. Motivated by [10, 25], we firstly study the auxiliary equation

$$\begin{aligned}{} & {} -\text {div}(g^2_\kappa (u)\nabla u)+g_\kappa (u)g^\prime _\kappa (u)|\nabla u|^2 + V(x)u\nonumber \\{} & {} \quad =\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^2}\frac{H(u)}{|x-y|^\mu |y|^\beta }dy\Bigg ) h(u),~x\in \mathbb {R}^2, \end{aligned}$$

(4.1)

where the continuous even function $g_\kappa :\mathbb {R}\rightarrow \mathbb {R}$ is defined by

$$\begin{aligned} g_\kappa (t)= \left\{ \begin{array}{ll} \sqrt{1-\kappa t^2}, &{}\quad \text {if}~0\le t<1/\sqrt{3\kappa },\\ \frac{1}{3\sqrt{2\kappa }t}+\frac{1}{\sqrt{6}}, &{}\quad \text {if}~ t\ge 1/\sqrt{3\kappa }, \\ g_\kappa (-t), &{}\quad \text {if}~t<0. \end{array} \right. \end{aligned}$$

Observe that if we get a solution u of Eq. (4.1) satisfying $|u|_\infty <1/\sqrt{3\kappa }$, then u is indeed a solution of the original Eq. (1.1). Consequently, to find a nontrivial solution with the prescribed $L^\infty$-estimate we shall make full use of the mountain-pass theorem together with the Mash-Moser iteration method. To our best knowledge, to exploit the Mash-Moser iteration successfully, we must take some more delicate analysis than that of in [25] because of the appearance of the Stein–Weiss convolution type nonlinearity with critical exponential growth.

From now on, throughout this section we rewrite $g_\kappa$ by g just for simplicity. Proceeding as [10, 25], we can present some properties on g below.

Lemma 4.1

For all $\kappa >0$, then

$(g_1)$:

$g\in C^1$ is increasing on $(-\infty ,0)$ and decreasing on $(0,+\infty )$ as well as

$$\begin{aligned} 1/\sqrt{6}\le g(t)\le 1~\text {and}~|g^\prime (t)|\le \sqrt{\kappa /2},~\forall t\in \mathbb {R}; \end{aligned}$$

$(g_2)$:

$-\frac{1}{2}\le t\frac{g^\prime (t)}{g(t)}\le 0$, for all $t\in \mathbb {R}$;

$(g_3)$:

The primitive $G(t)=\int _0^tg(\tau )d\tau$ of g(t) is an increasing function and therefore inevitable;

$(g_4)$:

$t\le G^{-1}(t)\le \sqrt{6}t$ for all $t\ge 0$, $\displaystyle \lim _{t\rightarrow 0}\frac{G^{-1}(t)}{t}=1$ and $\displaystyle \lim _{t\rightarrow +\infty }\frac{G^{-1}(t)}{t}=\sqrt{6}$.

Generally speaking, the (weak) solutions of Eq. (4.1) are critical points of its corresponding energy functional $I_\kappa :E\rightarrow \mathbb {R}$

$$\begin{aligned} I_\kappa (u)= & {} \frac{1}{2}\int _{\mathbb {R}^2}g^2(u)|\nabla u|^2dx+\frac{1}{2}\int _{\mathbb {R}^2}V(x)|u|^2dx\\{} & {} -\frac{1}{2} \int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(u(x)) H(u(y))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx. \end{aligned}$$

However, $I_\kappa$ may be not well-defined in E. With Lemma 4.1, we can get across it.

Let $u=G^{-1}(v)$, then we introduce the energy functional $\mathcal {J}_g=I_\kappa \circ G^{-1}:E\rightarrow \mathbb {R}$ defined by

$$\begin{aligned} \mathcal {J}_g(v)= & {} \frac{1}{2}\int _{\mathbb {R}^2}[|\nabla v|^2+V(x)|G^{-1}(v)|^2]dx\\{} & {} -\frac{1}{2}\int _{\mathbb {R}^2}\Bigg [|x|^{-\mu }*\big (|x|^{-\beta } H(G^{-1}(v)\big )\Bigg ]|x|^{-\beta } H(G^{-1}(v)dx. \end{aligned}$$

In fact, the critical points of $\mathcal {J}_g$ are solutions of

$$\begin{aligned}{} & {} -\Delta v+V(x) \frac{G^{-1}(v)}{g(G^{-1}(v))}\nonumber \\{} & {} \quad =\frac{1}{|x|^\beta }\Bigg (\int _{\mathbb {R}^2}\frac{H(G^{-1}(v))}{|x-y|^\mu |y|^\beta }dy\Bigg ) \frac{ h(v)}{g(G^{-1}(v))},~x\in \mathbb {R}^2. \end{aligned}$$

(4.2)

Thanks to the discussions in [10, 25], we conclude that $v\in E$ is a solution of Eq. (4.2) if and only if $u=G^{-1}(v)$ is a solution of Eq. (4.1).

Next, we would formulate the functional setting for a variational approach to Eq. (4.2). Because the nonlinearity f satisfies (1.9) and $(h_1)$, for fixed $\alpha >\alpha _0$, $q>1$ and for every $\varepsilon >0$, we have

$$\begin{aligned} |h(s)|\le \varepsilon |s|^{\frac{2-\mu }{2}} +C(\alpha ,q,\varepsilon )|s|^{q-1}(e^{\alpha s^2}-1),~\forall s\in \mathbb {R}\end{aligned}$$

(4.3)

and

$$\begin{aligned} |H(s)|\le \varepsilon |s|^{\frac{4-\mu }{2}} +C(\alpha ,q,\varepsilon )|s|^{q}(e^{\alpha s^2}-1),~\forall s\in \mathbb {R}. \end{aligned}$$

(4.4)

Given a function $v\in E$, by (1.3), we utilized $(g_4)$ and (4.4) with $\alpha >\alpha _0$ and $q\ge \frac{4-\mu }{2\nu }$ to obtain

$$\begin{aligned} \begin{aligned}&\Bigg |\int _{\mathbb {R}^2}\int _{\mathbb {R}^2}\frac{H(G^{-1}(v(x)))H(G^{-1}(v(y)))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx\Bigg | \le C_{\mu }\Bigg (\int _{\mathbb {R}^2}\frac{|H(G^{-1}(v))|^{\frac{4}{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{2}}\\&\quad \le C_{\mu }\Bigg (\int _{\mathbb {R}^2}\frac{ |v|^2}{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{2}}+C \Bigg (\int _{\mathbb {R}^2}|x|^{-\frac{4\beta }{4-\mu }}|v|^{\frac{4q}{4-\mu }}(e^{ \frac{24\alpha }{4-\mu }|v|^2}-1)dx\Bigg )^{\frac{4-\mu }{2}}\\&\quad \le C_{\beta ,\mu } \Vert v\Vert ^{4 -\mu }+C \Bigg (\int _{\mathbb {R}^2}|x|^{-\frac{4\beta }{4-\mu }}|v|^{\frac{4q\nu }{4 -\mu }}dx\Bigg )^{\frac{4 -\mu }{2\nu }}\\&\qquad \Bigg (\int _{\mathbb {R}^2}|x|^{-\frac{4\beta }{4-\mu }} \left( e^{ \frac{24\alpha \nu ^\prime }{4-\mu } |v|^2 }-1\right) dx\Bigg )^{\frac{4-\mu }{2\nu ^\prime }}\\&\quad \le C_{\beta ,\mu } \Vert v\Vert ^{4 -\mu }+C \Vert v \Vert ^{2q}\Bigg (\int _{\mathbb {R}^2}|x|^{-\frac{4\beta }{4-\mu }}\left( e^{ \frac{24\alpha \nu ^\prime }{4 -\mu } |v|^2 }-1\right) dx\Bigg )^{\frac{4 -\mu }{2\nu ^\prime }}<+\infty , \end{aligned} \end{aligned}$$

(4.5)

where we have adopted (1.10) in Proposition 1.2 together with $\nu >1$ and ${1}/{\nu }+{1}/{\nu ^\prime }=1$. Consequently, the energy functional $\mathcal {J}_g$ is well-defined and J is of class $\mathcal {C}^1$. Moreover, for all $\psi \in C_0^\infty (\mathbb {R}^2)$,

$$\begin{aligned} \mathcal {J}^\prime _g(v)[\psi ]= & {} \int _{\mathbb {R}^2}\Bigg [ \nabla v\nabla \psi +V(x) \frac{G^{-1}(v)\psi }{g(G^{-1}(v))} \Bigg ]dx\\{} & {} - \int _{\mathbb {R}^2}\Bigg [|x|^{-\mu }*\big (|x|^{-\beta } H(G^{-1}(v))\big )\Bigg ]\frac{h(G^{-1}(v))\psi }{g(G^{-1}(v))|x|^\beta }dx. \end{aligned}$$

To look for the mountain-pass type solutions for Eq. (4.2), firstly, we have to establish the validity of the mountain-pass geometry required in Proposition 2.2 for $\mathcal {J}_g$.

Lemma 4.2

Let (V) and (1.9) with $(h_1)$, $(h_6)-(h_7)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, then the functional $\mathcal {J}_g$ admits the following properties

(i):

there exist two constants $\varrho _\kappa ,\rho _\kappa >0$ such that $J_g(v)\ge \varrho _\kappa$ for all $v\in E$ with $\Vert v\Vert =\rho _\kappa$;

(ii):

there exists a function $v_1\in E$ with $\Vert v_1\Vert \ge \rho _\kappa$ such that $\mathcal {J}_g(v_1)<0$.

Proof

(i). Recalling (4.5), given a $v\in E$ with $\Vert v\Vert ^2<\frac{\pi (4-2\beta -\mu )}{6\alpha _0}$, we use $(g_4)$ and (1.11) to obtain

$$\begin{aligned} \mathcal {J}_g(v)\ge \frac{1}{2} \Vert v \Vert ^{2}- C_{\beta ,\mu } \Vert v\Vert ^{4 -\mu }-C \Vert v \Vert ^{2q}. \end{aligned}$$

Since $4-\mu >2$ and $q>1$, we can choose a suitably small $\rho _\kappa ^2<\frac{\pi (4-2\beta -\mu )}{6\alpha _0}$ to get the Point (i).

(ii). Let $\psi \in C^\infty _0(\mathbb {R}^2)$ with $|\psi |_\infty \le 1$, by $(h_5)$ and $(g_4)$,

$$\begin{aligned} \mathcal {J}_g(t\psi )\le & {} 3t^2\Vert \psi \Vert ^2-\frac{1}{2}\overline{\xi }^2t^{2\overline{p}} \int _{\mathbb {R}^2}\big [|x|^{-\mu }*\big (|x|^{-\beta } |\psi |^{\overline{p}}\big )\big ]|x|^{-\beta } |\psi |^{\overline{p}}dx\\\rightarrow & {} -\infty ~\text {as}~t\rightarrow +\infty \end{aligned}$$

since $\overline{p}>1$. Letting $v_1=t_0\psi$ with $t_0>0$ large enough, we can finish the proof of this lemma. $\square$

Via Proposition 2.2 and Lemma 4.2, there is a (C) sequence $\{v_n\}\subset E$ for $\mathcal {J}_g$ at the level $c_\kappa >0$. Although the mountain-pass value $c_\kappa$ is dependent of $\kappa >0$, we can take an upper bound for it.

Lemma 4.3

Let (V) and (1.9) with $(h_1)$, $(h_6)-(h_7)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, then there is a $\overline{\xi }_0>0$ such that for every $\overline{\xi }>\overline{\xi }_0$, there exists a $\overline{c}=\overline{c}(\overline{\xi })>0$ satisfying $c_\kappa \le \overline{c}<\frac{\pi (4-2\beta -\mu )(\overline{\eta }-1)}{12\alpha _0\overline{\eta }}$.

Proof

Let $\varphi _0\in C_0^\infty (\mathbb {R}^2)$ be a cutoff function used in Lemma 3.5. By $(g_4)$ and $(h_6)$,

$$\begin{aligned} \mathcal {J}_g(\varphi _0)&\le 3\int _{B_1(0)}[|\nabla \varphi _0|^2+V(x)|\varphi _0|^2]dx\nonumber \\ {}&\quad -\frac{1}{2}\int _{B_1(0)}\int _{B_1(0)} \frac{H(G^{-1}(\varphi _0(x))) H(G^{-1}(\varphi _0(x)))}{|x|^\beta |x-y|^\mu |y|^\beta }dydx \nonumber \\&< 3[1+ \max _{x\in B_1(0)}V(x)]\pi -2^{\mu -1}\overline{\xi }_1^2\Bigg (\int _{B_{1/2}(0)}|\varphi _0|^{2\overline{p}} dx\Bigg )^2\nonumber \\&\le 3[1+ \max _{x\in B_1(0)}V(x)]\pi -2^{\mu -3}\overline{\xi }_1^2\pi ^2=0. \end{aligned}$$

(4.6)

In particular, one can deduce from (4.6) that

$$\begin{aligned} \frac{1}{2}\int _{B_1(0)}[|\nabla \varphi _0|^2+V(x)|G^{-1}(\varphi _0)|^2]dx <2^{\mu -1}\overline{\xi }_1^2\Bigg (\int _{B_{1/2}(0)}|\varphi _0|^{2\overline{p}} dx\Bigg )^2. \end{aligned}$$

(4.7)

Choosing $\gamma _0(t)=t\varphi _0$, one easily deduces that $\gamma _0\in \Gamma _\kappa =\{\gamma \in C([0,1],E):\gamma (0)=0,\mathcal {J}_g(\gamma (1))<0\}$. According to the definition of $c_k$, then by (4.7) $(g_4)$ and $(h_6)$, we have

$$\begin{aligned} c_\kappa&\le \max _{t\in [0,1]}\mathcal {J}_g(t\varphi _0) \le \max _{t\ge 0}\Bigg \{ 3t^2 \int _{B_1(0)}[|\nabla \varphi _0|^2+V(x)|\varphi _0|^2]dx\\&-2^{\mu -1}\overline{\xi }^2 t^{2\overline{p}}\Bigg (\int _{B_{1/2}(0)}|\varphi _0|^{2\overline{p}} dx\Bigg )^2 \Bigg \} \\&\le \max _{t\ge 0}\Bigg \{3[1+ \max _{x\in B_1(0)}V(x)]\pi t^2-2^{\mu -5}\pi ^2\overline{\xi }^2 t^{2\overline{p}}\Bigg \} \\&=\frac{3(\overline{p}-1)[1+ \max _{x\in B_1(0)}V(x)]\pi }{\overline{p}}\Bigg (\frac{3[1+ \max _{x\in B_1(0)}V(x)]}{2^{\mu -5}\overline{p}\pi \overline{\xi }^2}\Bigg )^{\frac{1}{\overline{p}-1}}\triangleq \overline{c}. \end{aligned}$$

By using the assumption on $\xi$, there holds

$$\begin{aligned} \overline{\xi }^{-\frac{2}{\overline{p}-1}}<\frac{\pi (4-2\beta -\mu )(\overline{\eta }-1)\overline{p}}{36\alpha _0\overline{\eta } (\overline{p}-1)[1+ \max _{x\in B_1(0)}V(x)]\pi } \Bigg (\frac{2^{\mu -5}\overline{p}\pi }{3[1+ \max _{x\in B_1(0)}V(x)] }\Bigg )^{\frac{1}{\overline{p}-1}}. \end{aligned}$$

So, we can accomplish the proof of this lemma. $\square$

Now, with the help of Lemma 4.3, we can obtain the following result.

Lemma 4.4

Let (V) and (1.9) with $(h_1)$, $(h_6)-(h_7)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, then any $(C)_{c_\kappa }$ sequence of $\mathcal {J}_g$ is uniformly bounded.

Proof

Let $\{v_n\}\subset E$ be a $(C)_{c_\kappa }$ sequence of $\mathcal {J}_g$, namely $\mathcal {J}_g(v_n)\rightarrow c_\kappa$ and $(1+\Vert v_n\Vert )\Vert \mathcal {J}^\prime _g(v_n)\Vert _{E^{-1}}\rightarrow 0$ as $n\rightarrow \infty$. Inspired by [10, 25], we choose $\psi _n=G^{-1}(v_n)g(G^{-1}(v_n))$. By (V) and Lemma 4.1,

$$\begin{aligned} \int _{\mathbb {R}^2}V(x)|\psi _n|^2dx\le \int _{\mathbb {R}^2}V(x)|G^{-1}(v_n)|^2dx\le 6\int _{\mathbb {R}^2}V(x)|v_n|^2dx<+\infty \end{aligned}$$

and

$$\begin{aligned} \int _{\mathbb {R}^2} |\nabla \psi _n|^2dx= & {} \int _{\mathbb {R}^2}\Bigg [1+ G^{-1}(v_n)\frac{g^\prime (G^{-1}(v_n))}{g(G^{-1}(v_n))} \Bigg ]^2 |\nabla v_n|^2dx \\\le & {} \int _{\mathbb {R}^2} |\nabla v_n|^2dx<+\infty \end{aligned}$$

showing that $\{\psi _n\}\subset E$ and so $\psi _n$ could be a candidate for the test function.

Combining $(g_4)$ and $(h_7)$, it is simple to see that

$$\begin{aligned} c_\kappa +o_n(1)\Vert v_n\Vert&= \mathcal {J}_g(v_n)-\frac{1}{2\overline{\eta }}\mathcal {J}^\prime _g(v_n)[\psi _n] \nonumber \\&\ge \frac{\overline{\eta }-1}{2\overline{\eta }}\int _{\mathbb {R}^2}[|\nabla v_n|^2+V(x)| v_n|^2]dx\nonumber \\&-\frac{1}{2\eta }\int _{\mathbb {R}^2} \frac{G^{-1}(v_n)g^\prime {(G^{-1}(v_n))}}{g(G^{-1}(v_n))}|\nabla v_n|^2dx\nonumber \\&\ \ \ \ +\frac{1}{2\overline{\eta }}\int _{\mathbb {R}^2}\Bigg [|x|^{-\mu }*\big (|x|^{-\beta } H(G^{-1}(v_n)\big )\Bigg ] |x|^{-\beta } [h(G^{-1}(v_n))G^{-1}(v_n)\nonumber \\&- \eta H(G^{-1}(v_n)]dx\nonumber \\&\ge \frac{\overline{\eta }-1}{2\overline{\eta }}\Vert v_n\Vert ^2 \end{aligned}$$

(4.8)

which together with Lemma 4.3 implies the desired result. The proof is complete. $\square$

As a byproduct of Lemma 4.4, up to a subsequence if necessary, there is a function $v_\kappa \in E$ such that $v_n\rightharpoonup v_\kappa$ in E, $v_n\rightarrow v_\kappa$ in $L^p(\mathbb {R}^2,|x|^{-s}dx)$ for all $s\in (0,2)$ with $p\ge 2$ and $v_n\rightarrow v_\kappa$ a.e. in $\mathbb {R}^2$.

Next, we show that the energy functional $J_g$ satisfies the so-called $(C)_{c_\kappa }$ condition.

Lemma 4.5

Let (V) and (1.9) with $(h_1)$, $(h_6)-(h_7)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, then $\{v_n\}$ contains a strongly convergent subsequence for all fixed $\kappa >0$.

Proof

Due to $(h_1)$, without loss of generality, we can suppose that $v_n\ge 0$ in $\mathbb {R}^2$. Since $\{v_n\}\subset E$ is a $(C)_{c_\kappa }$ sequence of $\mathcal {J}_g$, combining Lemma 4.3 and (4.4), we could obtain $\displaystyle \limsup _{n\rightarrow \infty }\Vert v_n\Vert ^2<\frac{\pi (4-2\beta -\mu )}{6\alpha _0}$. Then, letting $\varepsilon =1$ in (4.3) and (4.4), we choose $\alpha >\alpha _0$ sufficiently close to $\alpha _0$ and $\nu ^\prime >1$ sufficiently close to 1 in such a way that $1/v+1/v^\prime =1$ and

$$\begin{aligned} \frac{\frac{24\alpha \nu ^\prime \Vert v_n\Vert ^2}{4-\mu }}{4\pi }+\frac{\frac{4\beta }{4-\mu }}{2}<1 \end{aligned}$$

(4.9)

for all large $n\in \mathbb {N}$. Hence, we apply (4.3), and (4.9) to get

$$\begin{aligned}{} & {} \int _{\mathbb {R}^2}\frac{|H(\sqrt{6}v_n)|^{\frac{4}{4-\mu }} }{|x|^{\frac{4\beta }{4-\mu }}}dx \le C\int _{\mathbb {R}^2}\frac{|v_n|^{2} }{|x|^{\frac{4\beta }{4-\mu }}}dx +C \int _{\mathbb {R}^2} \frac{|v_n|^{\frac{4q}{4-\mu }} \left( e^{\frac{24\alpha }{4-\mu } v_n^2}-1\right) }{|x|^{\frac{4\beta }{4-\mu }}}dx\\{} & {} \quad \le C\Vert v_n\Vert ^2+ C\Bigg (\int _{\mathbb {R}^2} \frac{|v_n|^{\frac{4q\nu }{4-\mu }} }{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{1}{\nu }} \Bigg (\int _{\mathbb {R}^2} \frac{\left( e^{\frac{24\alpha \nu ^\prime (1+\epsilon )^2|\nabla v_n|_2^2}{4-\mu } \left( \frac{v_n}{(1+\epsilon )|\nabla v_n|_2}\right) ^2 }-1\right) }{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{1}{\nu ^\prime }}\\{} & {} \quad \le C\Vert v_n\Vert ^2+ C\Vert v_n\Vert ^{\frac{4q}{4-\mu }}\le C<+\infty , \end{aligned}$$

where $\epsilon \approx 0^+$, $q=\nu \ge \frac{4-\mu }{2}$ and Lemma 2.3 are used. Note that $1/q+1/\nu ^\prime =1$, by means of the Hölder’s inequality,

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^2}\frac{|h(\sqrt{6}v_n)(v_n-v_\kappa )|^{\frac{4}{4-\mu }} }{|x|^{\frac{4\beta }{4-\mu }}}dx \\&\quad \le \int _{\mathbb {R}^2} \frac{|v_n|^{\frac{4(q-1)}{4-\mu }}|v_n-v_\kappa |^{\frac{4}{4-\mu }} \left( e^{\frac{24\alpha }{4-\mu } v_n^2}-1\right) }{|x|^{\frac{4\beta }{4-\mu }}}dx +C\int _{\mathbb {R}^2}\frac{|v_n|^{\frac{2(2-\mu )}{4-\mu }}|v_n-v_\kappa |^{\frac{4}{4-\mu }} }{|x|^{\frac{4\beta }{4-\mu }}}dx\\&\quad \le \Bigg (\int _{\mathbb {R}^2} \frac{|v_n|^{\frac{8q(q-1)}{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{1}{2q}} \Bigg (\int _{\mathbb {R}^2} \frac{|v_n-v_\kappa |^{\frac{8q}{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{1}{2q}} \Bigg (\int _{\mathbb {R}^2} \frac{\left( e^{\frac{24\alpha \nu ^\prime }{4-\mu } v_n^2}-1\right) }{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{1}{\nu ^\prime }}\\&\quad + C\Bigg (\int _{\mathbb {R}^2}\frac{|v_n|^{2} }{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^\frac{ 2-\mu }{4-\mu } \Bigg (\int _{\mathbb {R}^2}\frac{|v_n-v_\kappa |^{2} }{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^\frac{ 2 }{4-\mu }. \end{aligned} \end{aligned}$$

Recalling Proposition 1.2 and Lemma 2.3, one admits $v_n\rightarrow v_\kappa$ in $L^{2}(\mathbb {R}^2,|x|^{ -\frac{4\beta }{4-\mu } }dx)$ and $L^{\frac{8q}{4-\mu }}(\mathbb {R}^2,|x|^{ -\frac{4\beta }{4-\mu } }dx)$. It follows from the above two formulas as well as (4.9) that

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^2}\Bigg [|x|^{-\mu }*\big (|x|^{-\beta } H(G^{-1}(v_n))\big )\Bigg ]\frac{h(G^{-1}(v_n))(v_n-v_\kappa )}{g(G^{-1}(v_n))|x|^\beta }dx\\&\quad \le C_\mu \Bigg ( \int _{\mathbb {R}^2}\frac{|H(\sqrt{6}v_n)|^{\frac{4}{4-\mu }} }{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{4}} \Bigg ( \int _{\mathbb {R}^2}\frac{|h(\sqrt{6}v_n)(v_n-v_\kappa )|^{\frac{4}{4-\mu }} }{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{4}}\rightarrow 0. \end{aligned} \nonumber \\ \end{aligned}$$

(4.10)

Similarly, we can derive

$$\begin{aligned} \int _{\mathbb {R}^2}\Bigg [|x|^{-\mu }*\big (|x|^{-\beta } H(G^{-1}(v_\kappa ))\big )\Bigg ]\frac{h(G^{-1}(v_\kappa ))(v_n-v_\kappa )}{g(G^{-1}(v_\kappa ))|x|^\beta }dx\rightarrow 0. \end{aligned}$$

(4.11)

Let us recall that $(1+\Vert v_n\Vert )\Vert \mathcal {J}^\prime _g(v_n)\Vert _{E^{-1}}\rightarrow 0$, then invoking (4.10)–(4.11) and $(g_1)-(g_2)$, one could find that

$$\begin{aligned} o_n(1)&= \mathcal {J}^\prime _g(v_n)[v_n-v_\kappa ]-\mathcal {J}^\prime _g(v)[v_n-v_\kappa ] \\&=|\nabla v_n-\nabla v_\kappa |_2^2 +\int _{\mathbb {R}^2}V(x)\Bigg [ \frac{G^{-1}(v_n)}{g(G^{-1}(v_n))}- \frac{G^{-1}(v_\kappa )}{g(G^{-1}(v_\kappa ))}\Bigg ](v_n-v_\kappa )dx+o_n(1)\\&=|\nabla v_n-\nabla v_\kappa |_2^2 +\int _{\mathbb {R}^2}V(x)\frac{1}{g^2(G^{-1}(\vartheta _n))}\\&\quad \Bigg [1- \frac{G^{-1}(\vartheta _n)g^\prime (G^{-1}(\vartheta _n))}{g(G^{-1}(\vartheta _n))}\Bigg ]|v_n-v_\kappa |^2dx+o_n(1)\\&\ge \Vert v_n- v_\kappa \Vert ^2+o_n(1) \end{aligned}$$

showing that $v_n\rightarrow v_\kappa$ in E, where $\vartheta _n=\theta v_n+(1-\theta )v_\kappa$ is generated by the Mean Value theorem for some $\theta \in (0,1)$. So, the proof of this lemma is finished. $\square$

As a consequence, we can conclude that the weak limit $v_\kappa \in E$ is a nontrivial solution of Eq. (4.2).

Lemma 4.6

Let (V) and (1.9) with $(h_1)$, $(h_6)-(h_7)$ be satisfied. Assume that $\beta >0$, $0<\mu <2$ with $0<2\beta +\mu <2$, then Eq. (4.2) admits a nonnegative solution $v_\kappa \in E$ satisfying $\Vert v_\kappa \Vert ^2\le \frac{2\overline{\eta }}{\overline{\eta }-1}c_\kappa$.

Proof

By Proposition 2.2 and Lemma 4.2, there exists a (C) sequence $\{v_n\}\subset E$ for $\mathcal {J}_g$ at the level $c_\kappa >0$ (see Lemma 4.2-(i)). According to Lemmas 4.3, 4.4 and 4.5, we obtain that $\mathcal {J}_g(v_\kappa )=c_\kappa >0$ and $\mathcal {J}_g^\prime (v_\kappa )=0$ which reveal that $v_\kappa$ is a nontrivial solution of Eq. (4.2). Proceeding as (4.8), the estimate on $\Vert v_\kappa \Vert$ is immediate. Thus, we finish the proof of this lemma. $\square$

So far, we could deduce that $u_\kappa = G^{-1}(v_\kappa )$ is a nontrivial solution of Eq. (4.1). Whereas, we must consider the $L^\infty$-estimate for $u_k$ and so it would perhaps be a nontrivial solution of the original Eq. (1.1). By exploiting $(g_4)$ in Lemma 4.1, it suffices to investigate the $L^\infty$-estimate for $v_k$. Arguing as [10, 25], we reach this aim by the Nash–Moser iteration technique.

Nevertheless, different from those nonlinearities explored in [10, 25], we have to face some additional difficulties caused by the Stein–Weiss convolution parts. Consequently, we need to verify the following result.

Lemma 4.7

Under the assumptions in Lemma 4.6 and let $v_\kappa \in E$ be a nonnegative solution of Eq. (4.2) established by Lemma 4.6, then there is a constant $C_0>0$ independent of $\kappa$ such that

$$\begin{aligned} 0\le W_\kappa (x)\triangleq |x|^{-\mu }*\big (|x|^{-\beta } H(G^{-1}(v_\kappa ))\big )\le C_0<+\infty . \end{aligned}$$

Proof

Recalling Lemma 4.5, thanks to [44, Proposition 1], up to a subsequence if necessary, there is a function $\varpi \in E$ such that $|v_n(x)|\le \varpi (x)$ and $|v_\kappa (x)|\le \varpi (x)$ a.e. in $\mathbb {R}^2$. So, let $\varepsilon =1$ in (4.4), we have that

$$\begin{aligned} W_\kappa (x)\le \int _{\mathbb {R}^2}\frac{|\varpi |^{\frac{4-\mu }{2}} +C |\varpi |^{q}(e^{6\alpha \varpi ^2}-1)}{|y|^\beta |x-y|^{\mu }}dy. \end{aligned}$$

To finish the proof, we show that the above integral is well-defined. In view of the Claim 1 in Lemma 2.9, we can find a constant $C_1^\varpi >0$ such that

$$\begin{aligned} \int _{\mathbb {R}^2}\frac{|\varpi |^{\frac{4-\mu }{2}} }{|y|^\beta |x-y|^{\mu }}dy\le C_1^\varpi . \end{aligned}$$

We continue to follow the proof of Claim 1 in Lemma 2.9 to have

$$\begin{aligned}&\int _{|x-y|\le 1}\frac{ |\varpi |^{q}(e^{6\alpha \varpi ^2}-1)}{|y|^\beta |x-y|^{\mu }}dy \le C\Bigg (\int _{\mathbb {R}^2} \frac{ |\varpi |^{\frac{4q}{2+\beta -\mu }}\left( e^{\frac{24\alpha }{2+\beta -\mu } \varpi ^2}-1\right) }{|y|^{\frac{4\beta }{2+\beta -\mu }}}dy \Bigg )^{\frac{2+\beta -\mu }{4}} \\&\quad \le C\Bigg (\int _{\mathbb {R}^2} \frac{ |\varpi |^{\frac{8q}{2+\beta -\mu }} }{|y|^{\frac{4\beta }{2+\beta -\mu }}}dy \Bigg )^{\frac{2+\beta -\mu }{8}} \Bigg (\int _{\mathbb {R}^2} \frac{ \left( e^{\frac{48\alpha }{2+\beta -\mu } \varpi ^2}-1\right) }{|y|^{\frac{4\beta }{2+\beta -\mu }}}dy \Bigg )^{\frac{2+\beta -\mu }{8}}<+\infty \end{aligned}$$

and since $|x-y|^{-\mu }\le 1$ whenever $|x-y|>1$

$$\begin{aligned} \int _{|x-y|>1}\frac{ |\varpi |^{q}(e^{6\alpha \varpi ^2}-1)}{|y|^\beta |x-y|^{\mu }}dy \le \Bigg (\int _{\mathbb {R}^2}\frac{ |\varpi |^{2q} }{|y|^\beta }dy\Bigg )^{\frac{1}{2}} \Bigg (\int _{\mathbb {R}^2}\frac{ (e^{12\alpha \varpi ^2}-1)}{|y|^\beta }dy\Bigg )^{\frac{1}{2}}<+\infty \end{aligned}$$

Thereby, there is a constant $C_2^\varpi >0$ such that

$$\begin{aligned} C\int _{\mathbb {R}^2}\frac{ |\varpi |^{q}(e^{6\alpha \varpi ^2}-1)}{|y|^\beta |x-y|^{\mu }}dy\le C_2^\varpi . \end{aligned}$$

Now, we can accomplish the proof of this lemma by choosing $C_0=C_1^\varpi +C_2^\varpi \in (0,+\infty )$. $\square$

Lemma 4.8

Under the assumptions in Lemma 4.6 and let $v_\kappa \in E$ be the nonnegative solution of Eq. (4.2) established by Lemma 4.6. If in addition $\lim _{s\rightarrow 0^+}\frac{h(s)}{s}=0$, then $v_\kappa \in L^\infty (\mathbb {R}^2)$ and

$$\begin{aligned} |v_\kappa |_\infty \le \Bigg (\frac{2}{\tilde{q}}\Bigg )^{\frac{2\tilde{q}}{(2-\tilde{q})^2}}\Bigg (\overline{C}\int _{\mathbb {R}^2}(e^{6\alpha \tilde{q}^\prime \varpi ^2}-1)dx \Bigg )^{\frac{\tilde{q}}{2\tilde{q}^\prime (2-\tilde{q})}}|v_\kappa |_4^4, \end{aligned}$$

where $\overline{C}>0$ is a constant independent of $\kappa$ and $\tilde{q}\in (1,2)$ with $\frac{1}{\tilde{q}}+\frac{1}{\tilde{q}^\prime }=1$.

Proof

Let $\gamma >1$ and $m\in \mathbb {N}^+$ and we define the sets $A_m\triangleq \{x\in \mathbb {R}^2:v_\kappa ^{\gamma -1}\le m\}$ and $B_m=\mathbb {R}^2\backslash A_m$. Consider the sequences

$$\begin{aligned} (v_\kappa )_m= \left\{ \begin{array}{ll} v_\kappa ^{2\gamma -1}, &{}\quad \text {in}~A_m, \\ m^2 v_\kappa , &{}\quad \text {in}~B_m, \end{array} \right. \text {and}~ (w_\kappa )_m= \left\{ \begin{array}{ll} v_\kappa ^\gamma , &{}\quad \text {in}~A_m, \\ m v_\kappa , &{}\quad \text {in}~B_m. \end{array} \right. \end{aligned}$$

It’s simple to see that $(v_\kappa )_m,(w_\kappa )_m\in E$, $|(v_\kappa )_m|\le |v_\kappa |^{2\gamma -1}$ and $|(w_\kappa )_m|^2=v_\kappa (v_\kappa )_m\le |v_\kappa |^{2\gamma }$ in $\mathbb {R}^2$. Moreover,

$$\begin{aligned} \nabla (v_\kappa )_m= \left\{ \begin{array}{ll} (2\gamma -1)v_\kappa ^{2(\gamma -1)}\nabla v_\kappa , &{}\quad \text {in}~A_m, \\ m^2\nabla v_\kappa , &{}\quad \text {in}~B_m, \end{array} \right. \text {and}~ \nabla (w_\kappa )_m= \left\{ \begin{array}{ll} \gamma v_\kappa ^{\gamma -1}\nabla v_\kappa , &{}\quad \text {in}~A_m, \\ m \nabla v_\kappa , &{}\quad \text {in}~B_m, \end{array} \right. \end{aligned}$$

which imply that

$$\begin{aligned} \left\{ \begin{array}{ll} \int _{\mathbb {R}^2} \nabla v_\kappa \nabla (v_\kappa )_mdx=(2\gamma -1) \int _{A_m} v_\kappa ^{2(\gamma -1)}|\nabla v_\kappa |^2dx+m^2\int _{B_m} |\nabla v_\kappa |^2dx, \\ \int _{\mathbb {R}^2}|\nabla (w_\kappa )_m|^2dx=\gamma ^2 \int _{A_m} v_\kappa ^{2(\gamma -1)}|\nabla v_\kappa |^2dx+m^2\int _{B_m} |\nabla v_\kappa |^2dx. \end{array} \right. \nonumber \\ \end{aligned}$$

(4.12)

Combining (4.12) and the fact that $\gamma >1$, one obtains

$$\begin{aligned} \int _{\mathbb {R}^2}|\nabla (w_\kappa )_m|^2dx&= \int _{\mathbb {R}^2} \nabla v_\kappa \nabla (v_\kappa )_mdx +(\gamma -1)^2 \int _{A_m} v_\kappa ^{2(\gamma -1)}|\nabla v_\kappa |^2dx\nonumber \\&\le \Bigg [1+\frac{(\gamma -1)^2}{2\gamma -1}\Bigg ]\int _{\mathbb {R}^2} \nabla v_\kappa \nabla (v_\kappa )_mdx \le \gamma ^2 \int _{\mathbb {R}^2} \nabla v_\kappa \nabla (v_\kappa )_mdx. \end{aligned}$$

(4.13)

Because $v_\kappa \in E$ is a nontrivial critical point of $\mathcal {J}_g$, that is, $\mathcal {J}^\prime _g(v_\kappa )[(v_\kappa )_m]=0$ which gives that

$$\begin{aligned} \int _{\mathbb {R}^2} \nabla v_\kappa \nabla (v_\kappa )_m+V(x) \frac{G^{-1}(v_\kappa )(v_\kappa )_m}{g(G^{-1}(v_\kappa ))} dx =\int _{\mathbb {R}^2}W_\kappa \frac{h(G^{-1}(v_\kappa ))(v_\kappa )_m}{g(G^{-1}(v_\kappa ))|x|^\beta }dx. \end{aligned}$$

By Lemma 4.7, we have

$$\begin{aligned}{} & {} \int _{\mathbb {R}^2} \nabla v_\kappa \nabla (v_\kappa )_m+V(x) \frac{G^{-1}(v_\kappa )(v_\kappa )_m}{g(G^{-1}(v_\kappa ))} dx \nonumber \\{} & {} \quad \le C_0\int _{\mathbb {R}^2} \frac{h(G^{-1}(v_\kappa ))(v_\kappa )_m}{g(G^{-1}(v_\kappa ))|x|^\beta }dx. \end{aligned}$$

(4.14)

Since $h(s)=o(s)$ as $s\rightarrow 0^+$, by (1.9), there is a constant $\overline{p}>1$ and $C_{\alpha ,\overline{p}}$ dependent of $\alpha$ and $\overline{p}$ such that

$$\begin{aligned} |h(s)|\le \frac{V_0}{2C_0} |s| +C_{\alpha ,\overline{p}}|s|^{\overline{p}-1}(e^{\alpha s^2}-1),~\forall s\in \mathbb {R}. \end{aligned}$$

(4.15)

Note that $v_\kappa (v_\kappa )_m=(w_\kappa )_m^2$ and $V(x)\ge V_0$ by (V), by using (4.13), (4.14) and (4.15) with $\overline{p}=2$ to get

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^2}|\nabla (w_\kappa )_m|^2dx \le \gamma ^2 \int _{\mathbb {R}^2} \nabla v_\kappa \nabla (v_\kappa )_mdx\\&\quad \le \gamma ^2\Bigg (C_0\int _{\mathbb {R}^2}\frac{h(G^{-1}(v_\kappa ))(v_\kappa )_m}{g(G^{-1}(v_\kappa ))}dx - \int _{\mathbb {R}^2} V(x)\frac{G^{-1}(v_\kappa )(v_\kappa )_m}{g(G^{-1}(v_\kappa ))} dx\Bigg ) \\&\quad \le C_{\alpha ,2} C_0\gamma ^2\int _{\mathbb {R}^2}\frac{ G^{-1}(v_\kappa )(v_\kappa )_m}{g(G^{-1}(v_\kappa ))}[e^{\alpha |G^{-1}(v_\kappa )|^2}-1]dx\\&\qquad -\frac{1}{2}\int _{\mathbb {R}^2}V(x)\frac{ G^{-1}(v_\kappa )(v_\kappa )_m}{g(G^{-1}(v_\kappa ))}dx \end{aligned} \end{aligned}$$

which together with the facts $1/\sqrt{6}\le g(s)\le 1$ and $s\le G^{-1}(s)\le \sqrt{6}s$ for all $s\ge 0$ gives that

$$\begin{aligned}{} & {} \int _{\mathbb {R}^2}(|\nabla (w_\kappa )_m|^2+V(x)|(w_\kappa )_m|^2)dx \\{} & {} \quad \le 2\sqrt{6}C_{\alpha ,2} C_0\gamma ^2\int _{\mathbb {R}^2} (w_\kappa )_m^2[e^{6\alpha \varpi ^2}-1]dx,~\forall m\in \mathbb {N}. \end{aligned}$$

We fix $\tilde{q}\in (1,2)$ with $\tilde{q}^\prime =\widetilde{q}/(\widetilde{q}-1)$ and $E\hookrightarrow L^4(\mathbb {R}^2)$, there is a constant $\overline{C}>0$ independent $\gamma$ and $\kappa$ such that

$$\begin{aligned} \Bigg (\int _{\mathbb {R}^2} |(w_\kappa )_m|^4 dx\Bigg )^{\frac{1}{2}} \le \overline{C} \gamma ^2 I_{\alpha ,\tilde{q}^\prime }\Bigg (\int _{\mathbb {R}^2} |(w_\kappa )_m|^{2\tilde{q}} dx\Bigg )^{\frac{1}{\tilde{q}}},~\forall m\in \mathbb {N}, \end{aligned}$$

where

$$\begin{aligned} I_{\alpha ,\tilde{q}^\prime }\triangleq \Bigg (\int _{\mathbb {R}^2}(e^{6\alpha \tilde{q}^\prime \varpi ^2}-1)dx\Bigg )^{\frac{1}{\tilde{q}^\prime }}. \end{aligned}$$

Once $(w_\kappa )_m=v_\kappa ^\gamma$ in $A_m$ and $(w_\kappa )_m\le v_\kappa ^\gamma$ in $\mathbb {R}^2$, there holds

$$\begin{aligned} \Bigg (\int _{A_m} |v_\kappa |^{4\gamma } dx\Bigg )^{\frac{1}{2}} \le \overline{C} \gamma ^2 I_{\alpha ,\tilde{q}^\prime }\Bigg (\int _{\mathbb {R}^2} |v_\kappa |^{2\tilde{q}\gamma } dx\Bigg )^{\frac{1}{\tilde{q}}},~\forall m\in \mathbb {N}. \end{aligned}$$

By applying the Dominated Convergence theorem with $m\rightarrow \infty$ to the above formula, one has

$$\begin{aligned} |v_\kappa |_{4\gamma }^{2\gamma }\le \overline{C} \gamma ^2 I_{\alpha ,\tilde{q}^\prime }|v_\kappa |_{2\tilde{q}\gamma }^{2\gamma }. \end{aligned}$$

(4.16)

We choose the constant $\sigma =2/\tilde{q}$, then $\sigma >1$ because $\tilde{q}\in (1,2)$. For every $j\in \mathbb {N}^+$, define $\gamma _j=\sigma ^j$ and thus $2\tilde{q}\gamma _{j+1}=2\tilde{q}\sigma \gamma _j=4\gamma _j$. For $j=1$, $\gamma _1=\sigma >1$ which can be applied in (4.16) to derive

$$\begin{aligned} |v_\kappa |_{4\sigma } \le \sigma ^{\frac{1}{\sigma }} (\overline{C}I_{\alpha ,\tilde{q}^\prime })^{\frac{1}{2\sigma }}|v_\kappa |_{4}. \end{aligned}$$

(4.17)

For $j=2$, $\gamma _2=\sigma ^2>1$ and $2\tilde{q}\gamma _2=4\gamma _1=4\sigma$ and by (4.16),

$$\begin{aligned} |v_\kappa |_{4\sigma ^2}\le (\sigma ^2)^{\frac{1}{\sigma ^2}} (\overline{C}I_{\alpha ,\tilde{q}^\prime })^{\frac{1}{2\sigma ^2}}|v_\kappa |_{4\sigma }. \end{aligned}$$

(4.18)

For $j=3$, $\gamma _3=\sigma ^3>1$ and $2\tilde{q}\gamma _3=4\gamma _2=4\sigma ^2$ and by (4.16),

$$\begin{aligned} |v_\kappa |_{4\sigma ^3}\le (\sigma ^3)^{\frac{1}{\sigma ^3}} (\overline{C} I_{\alpha ,\tilde{q}^\prime })^{\frac{1}{2\sigma ^3}}|v_\kappa |_{4\sigma ^2}. \end{aligned}$$

(4.19)

Similar to (4.17), (4.18) and (4.19), proceeding this iteration procedure j times, we can infer that

$$\begin{aligned} |v_\kappa |_{4\sigma ^j}\le \sigma ^{\sum _{i=1}^j\frac{i}{\sigma ^i}} (\overline{C} I_{\alpha ,\tilde{q}^\prime })^{\frac{1}{2}\sum _{i=1}^j\frac{1}{\sigma ^i}}|v_\kappa |_{4} \end{aligned}$$

(4.20)

invoking that $v_\kappa \in L^{4\sigma ^j}(\mathbb {R}^2)$ for all $j\in \mathbb {N}^+$. Clearly, $\sum _{i=1}^\infty \frac{i}{\sigma ^i}=\frac{\sigma }{(\sigma -1)^2}$ and $\sum _{i=1}^\infty \frac{1}{\sigma ^i}=\frac{1}{\sigma -1}$, thereby we can take the limit in (4.20) as $j\rightarrow \infty$ to obtain

$$\begin{aligned} |v_\kappa |_\infty \le \sigma ^{\frac{\sigma }{(\sigma -1)^2}}(\overline{C} I_{\alpha ,\tilde{q}^\prime })^{\frac{1}{2(\sigma -1)}}|v_\kappa |_4^4 \end{aligned}$$

finishing the proof of this lemma. $\square$

At this step, we can present the proof of Theorem 1.8.

Proof of Theorem 1.8

Combining Lemmas 4.6 and 4.8, we know that Eq. (4.2) has a nonnegative solution $v_\kappa \in E$ satisfying

$$\begin{aligned} |v_\kappa |_\infty \le d_0\triangleq \frac{\pi (4-2\beta -\mu ) }{6\alpha _0 } \Bigg (\frac{2}{\tilde{q}}\Bigg )^{\frac{2\tilde{q}}{(2-\tilde{q})^2}}\Bigg (\overline{C}\int _{\mathbb {R}^2}(e^{6\alpha \tilde{q}^\prime \varpi ^2}-1)dx \Bigg )^{\frac{\tilde{q}}{2\tilde{q}^\prime (2-\tilde{q})}}, \end{aligned}$$

where $\varpi \in E$ is given by Lemma 4.7. So, we can define $\kappa _0=1/(18d_0^2)$ and then for all $\kappa \in (0,\kappa _0)$

$$\begin{aligned} |u_\kappa |_\infty =|G^{-1}(v_\kappa )|_\infty \le \sqrt{6}|v_\kappa |_\infty \le \sqrt{6}d_0 =\frac{1}{\sqrt{3\kappa _0}}<\frac{1}{\sqrt{3\kappa }}. \end{aligned}$$

(4.21)

In summary, $u_\kappa$ is a nontrivial nonnegative solution of Eq. (4.1). By (4.21), we recall the definition of $g=g_\kappa$ to conclude that $u_\kappa \in E\cap L^\infty (\mathbb {R}^2)$ is a nontrivial nonnegative solution of Eq. (1.1). $\square$

Finally, we study the asymptotical behavior of $u_\kappa$ as $\kappa \rightarrow 0^+$.

Proof of Theorem 1.11

Since $u_\kappa =G^{-1}(v_\kappa )$ is a solution of Eq. (1.1), via Lemmas 4.3 and 4.6,

$$\begin{aligned} \Vert u_\kappa \Vert ^2= & {} \int _{\mathbb {R}^2}\Bigg [ \frac{|\nabla v_\kappa |^2}{g^2(G^{-1}(v_\kappa ))} +V(x)|G^{-1}(v_\kappa )|^2 \Bigg ]dx\nonumber \\\le & {} 6\Vert v_\kappa \Vert ^2\le \frac{12\overline{\eta }}{\overline{\eta }-1}c_\kappa <\frac{\pi (4-2\beta -\mu ) }{ \alpha _0}. \end{aligned}$$

(4.22)

So, going to a subsequence if necessary, there is a function $u_0\in E$ such that $u_\kappa \rightharpoonup u_0$ in E, $u_\kappa \rightarrow u_0$ in $L^p(\mathbb {R}^2,|x|^{-s}dx)$ for each $p\in (2,+\infty )$ with $s\in (0,1)$, and $u_\kappa \rightarrow u_0$ a.e. in $\mathbb {R}^2$ as $\kappa \rightarrow 0^+$. Recalling that $u_\kappa$ is a solution of Eq. (1.1), for all $\psi \in C_0^\infty (\mathbb {R}^2)$, there holds

$$\begin{aligned} \begin{aligned}&\int _{\mathbb {R}^2}[ \nabla u_\kappa \nabla \psi +V(x)u_\kappa \psi ]dx+\kappa \int _{\mathbb {R}^2} (|\nabla u_\kappa |^2 u_\kappa \psi + |u_\kappa |^2 \nabla u_\kappa \nabla \psi )dx\\&\quad = \int _{\mathbb {R}^2} [|x|^{-\mu }*(|x|^{-\beta }H(u_\kappa )) ]|x|^{-\beta } h(u_\kappa )\psi dx.\\ \end{aligned} \nonumber \\ \end{aligned}$$

(4.23)

In view of $|u_\kappa |_\infty \le \sqrt{6}d_0$ in (4.21) and (4.22), as $\kappa \rightarrow 0^+$, we obtain

$$\begin{aligned}{} & {} \Bigg |\kappa \int _{\mathbb {R}^2} (|\nabla u_\kappa |^2 u_\kappa \psi + |u_\kappa |^2 \nabla u_\kappa \nabla \psi )dx\Bigg | \nonumber \\{} & {} \quad \le \kappa |u_\kappa |_\infty |\psi |_\infty |\nabla u_\kappa |^2_2 +\kappa |u_\kappa |_\infty ^2 |\nabla u_\kappa |_2|\nabla \psi |_2\rightarrow 0. \end{aligned}$$

(4.24)

In particular, choosing $\psi =u_\kappa -u_0\in E$ and so $|\psi |_\infty$ and $|\nabla \psi |_2$ are uniformly bounded in $\kappa \in (0,\kappa _0)$ by (4.21) and (4.22). Inserting $\psi =u_\kappa -u_0$ into (4.24), as $\kappa \rightarrow 0^+$, we have that

$$\begin{aligned} \Bigg |\kappa \int _{\mathbb {R}^2} [|\nabla u_\kappa |^2 u_\kappa (u_\kappa -u_0)+ |u_\kappa |^2 \nabla u_\kappa \nabla (u_\kappa -u_0)]dx\Bigg | \rightarrow 0. \end{aligned}$$

(4.25)

Next, we claim that,

$$\begin{aligned}{} & {} \int _{\mathbb {R}^2} [|x|^{-\mu }*(|x|^{-\beta }H(u_\kappa )) ]|x|^{-\beta } h(u_\kappa )u_\kappa dx \nonumber \\{} & {} \quad \rightarrow \int _{\mathbb {R}^2} [|x|^{-\mu }*(|x|^{-\beta }H(u_0)) ]|x|^{-\beta } h(u_0)u_0 dx \end{aligned}$$

(4.26)

and

$$\begin{aligned}{} & {} \int _{\mathbb {R}^2} [|x|^{-\mu }*(|x|^{-\beta }H(u_\kappa )) ]|x|^{-\beta } h(u_\kappa )\psi dx\nonumber \\{} & {} \quad \rightarrow \int _{\mathbb {R}^2} [|x|^{-\mu }*(|x|^{-\beta }H(u_0)) ]|x|^{-\beta } h(u_0)\psi dx. \end{aligned}$$

(4.27)

Indeed, we argue as the proof of Claim 1 in Lemma 2.9 to derive that $[|x|^{-\mu }*(|x|^{-\beta }H(u_\kappa )) ]|x|^{-\beta } h(u_\kappa )u_\kappa$ $\rightarrow [|x|^{-\mu }*(|x|^{-\beta }H(u_0)) ]|x|^{-\beta } h(u_0)u_0$ a.e. in $\mathbb {R}^2$. Therefore, letting $\varepsilon =1$ in (4.3), we choose $\alpha >\alpha _0$ sufficiently close to $\alpha _0$ and $\nu ^\prime >1$ sufficiently close to 1 in such a way that $1/v+1/v^\prime =1$ and

$$\begin{aligned} \frac{\frac{4\alpha \nu ^\prime \Vert u_\kappa \Vert ^2}{4-\mu }}{4\pi }+\frac{\frac{4\beta }{4-\mu }}{2}<1 \end{aligned}$$

(4.28)

for all sufficiently small $\kappa \in (0,\kappa _0)$. Using $(h_6)$, the HLS inequality, (4.3) and (1.11) with (4.28),

$$\begin{aligned}{} & {} \Bigg |\int _{\mathbb {R}^2} [|x|^{-\mu }*(|x|^{-\beta }H(u_\kappa )) ]|x|^{-\beta } h(u_\kappa )u_\kappa dx\Bigg |\le \overline{\eta } C_\mu \Bigg ( \int _{\mathbb {R}^2} \frac{|h(u_\kappa )u_\kappa |^{\frac{4}{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}}dx \Bigg )^{\frac{4-\mu }{2}}\nonumber \\{} & {} \quad \le C\Bigg ( \int _{\mathbb {R}^2}\frac{|u_\kappa |^{2} }{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{2}} +C\Bigg ( \int _{\mathbb {R}^2} \frac{|u_\kappa |^{\frac{4q}{4-\mu }} \left( e^{\frac{4\alpha }{4-\mu } u_\kappa ^2}-1\right) }{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{2}}\nonumber \\{} & {} \quad \le C\Bigg ( \int _{\mathbb {R}^2}\frac{|u_\kappa |^{2} }{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{2}} + C\Bigg (\int _{\mathbb {R}^2} \frac{|u_\kappa |^{\frac{4q\nu }{4-\mu }} }{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{1}{\nu }}\nonumber \\{} & {} \qquad \Bigg (\int _{\mathbb {R}^2} \frac{\left( e^{\frac{4\alpha \nu ^\prime (1+\epsilon )^2 |\nabla u_\kappa |_2^2}{4-\mu } \left( \frac{u_\kappa }{(1+\epsilon )|u_\kappa |_2}\right) ^2 }-1\right) }{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{1}{\nu ^\prime }}\nonumber \\{} & {} \quad \le C\Bigg ( \int _{\mathbb {R}^2}\frac{|u_\kappa |^{2} }{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{4-\mu }{2}} + C\Bigg (\int _{\mathbb {R}^2} \frac{|u_\kappa |^{\frac{4q\nu }{4-\mu }} }{|x|^{\frac{4\beta }{4-\mu }}}dx\Bigg )^{\frac{1}{\nu }} \end{aligned}$$

(4.29)

for $\epsilon \approx 0^+$, which indicates (4.26) by Lemma 2.3 and the generalized Lebesgue’s Dominated Convergence theorem. One can get (4.27) in a very similar way. We tend $\kappa \rightarrow 0^+$ in (4.23) as well as with (4.24) and (4.27) to derive

$$\begin{aligned} \int _{\mathbb {R}^2}[ \nabla u_0 \nabla \psi +V(x)u_0\psi ]dx = \int _{\mathbb {R}^2} [|x|^{-\mu }*(|x|^{-\beta }H(u_0)) ]|x|^{-\beta } h(u_0)\psi dx \end{aligned}$$

showing that $u_0\in E$ is a solution of Eq. (1.17). Moreover, with the help of (4.23) for $\psi =u_\kappa -u_0$, one would conclude that $u_\kappa \rightarrow u_0$ in E by (4.24) and (4.26)–(4.27). Eventually, to accomplish the proof, we have to verify that $u_0\ne 0$. To the end, letting $\psi =u_\kappa$ in (4.23), by (4.29) with $q=\frac{(4-\mu )^2}{4}$,

$$\begin{aligned} \Vert u_\kappa \Vert ^2&\le \int _{\mathbb {R}^2} [|x|^{-\mu }*(|x|^{-\beta }H(u_\kappa )) ]|x|^{-\beta } h(u_\kappa )u_\kappa dx \\&\le \overline{\eta } C_\mu \Bigg ( \int _{\mathbb {R}^2} \frac{|h(u_\kappa )u_\kappa |^{\frac{4}{4-\mu }}}{|x|^{\frac{4\beta }{4-\mu }}}dx \Bigg )^{\frac{4-\mu }{2}} \\&\le C \Vert u_\kappa \Vert ^{4-\mu }+C \Vert u_\kappa \Vert ^{\frac{4q}{4-\mu }}=C \Vert u_\kappa \Vert ^{4-\mu } \end{aligned}$$

yielding that $\Vert u_\kappa \Vert \ge C>0$ since $\mu <2$, where $C>0$ is independent of $\kappa \in (0,\kappa _0)$. Therefore, we have that $\Vert u_0\Vert \ge C>0$ by $u_\kappa \rightarrow u_0$ in E. The proof is finished. $\square$